id	sid	tid	token	lemma	pos
asir-4475	1	1	applied	apply	VERB
asir-4475	1	2	science	science	NOUN
asir-4475	1	3	and	and	CCONJ
asir-4475	1	4	innovative	innovative	ADJ
asir-4475	1	5	research	research	NOUN
asir-4475	1	6	issn	issn	VERB
asir-4475	1	7	2474	2474	NUM
asir-4475	1	8	-	-	SYM
asir-4475	1	9	4972	4972	NUM
asir-4475	1	10	(	(	PUNCT
asir-4475	1	11	print	print	NOUN
asir-4475	1	12	)	)	PUNCT
asir-4475	1	13	issn	issn	VERB
asir-4475	1	14	2474	2474	NUM
asir-4475	1	15	-	-	SYM
asir-4475	1	16	4980	4980	NUM
asir-4475	1	17	(	(	PUNCT
asir-4475	1	18	online	online	ADJ
asir-4475	1	19	)	)	PUNCT
asir-4475	1	20	vol	vol	NOUN
asir-4475	1	21	.	.	PROPN
asir-4475	2	1	6	6	NUM
asir-4475	2	2	,	,	PUNCT
asir-4475	2	3	no	no	INTJ
asir-4475	2	4	.	.	NOUN
asir-4475	2	5	1	1	NUM
asir-4475	2	6	,	,	PUNCT
asir-4475	2	7	2022	2022	NUM
asir-4475	2	8	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	VERB
asir-4475	2	9	14	14	NUM
asir-4475	2	10	original	original	ADJ
asir-4475	2	11	paper	paper	NOUN
asir-4475	2	12	riemann	riemann	PROPN
asir-4475	2	13	’s	’s	PART
asir-4475	2	14	hypothesis	hypothesis	NOUN
asir-4475	2	15	new	new	ADJ
asir-4475	2	16	proof	proof	ADJ
asir-4475	2	17	fausto	fausto	PROPN
asir-4475	2	18	galetto1	galetto1	PROPN
asir-4475	2	19	1	1	NUM
asir-4475	2	20	independent	independent	ADJ
asir-4475	2	21	researcher	researcher	NOUN
asir-4475	2	22	,	,	PUNCT
asir-4475	2	23	past	past	ADP
asir-4475	2	24	professor	professor	NOUN
asir-4475	2	25	of	of	ADP
asir-4475	2	26	quality	quality	NOUN
asir-4475	2	27	management	management	NOUN
asir-4475	2	28	at	at	ADP
asir-4475	2	29	politecnico	politecnico	PROPN
asir-4475	2	30	of	of	ADP
asir-4475	2	31	turin	turin	PROPN
asir-4475	2	32	,	,	PUNCT
asir-4475	2	33	turin	turin	PROPN
asir-4475	2	34	,	,	PUNCT
asir-4475	2	35	italy	italy	PROPN
asir-4475	2	36	received	receive	VERB
asir-4475	2	37	:	:	PUNCT
asir-4475	2	38	december	december	PROPN
asir-4475	2	39	28	28	NUM
asir-4475	2	40	,	,	PUNCT
asir-4475	2	41	2021	2021	NUM
asir-4475	2	42	accepted	accept	VERB
asir-4475	2	43	:	:	PUNCT
asir-4475	2	44	january	january	PROPN
asir-4475	2	45	12	12	NUM
asir-4475	2	46	,	,	PUNCT
asir-4475	2	47	2022	2022	NUM
asir-4475	2	48	online	online	ADV
asir-4475	2	49	published	publish	VERB
asir-4475	2	50	:	:	PUNCT
asir-4475	2	51	january	january	PROPN
asir-4475	2	52	19	19	NUM
asir-4475	2	53	,	,	PUNCT
asir-4475	2	54	2022	2022	NUM
asir-4475	2	55	doi:10.22158	doi:10.22158	NOUN
asir-4475	2	56	/	/	SYM
asir-4475	2	57	asir.v6n1p14	asir.v6n1p14	NOUN
asir-4475	2	58	url	url	PROPN
asir-4475	2	59	:	:	PUNCT
asir-4475	2	60	http://doi.org/10.22158/asir.v6n1p14	http://doi.org/10.22158/asir.v6n1p14	NOUN
asir-4475	2	61	abstract	abstract	ADV
asir-4475	2	62	after	after	SCONJ
asir-4475	2	63	some	some	DET
asir-4475	2	64	papers	paper	NOUN
asir-4475	2	65	proving	prove	VERB
asir-4475	2	66	the	the	DET
asir-4475	2	67	rc	rc	PROPN
asir-4475	2	68	(	(	PUNCT
asir-4475	2	69	short	short	ADJ
asir-4475	2	70	for	for	ADP
asir-4475	2	71	“	"	PUNCT
asir-4475	2	72	riemann	riemann	PROPN
asir-4475	2	73	’s	’s	PART
asir-4475	2	74	conjecture	conjecture	NOUN
asir-4475	2	75	”	"	PUNCT
asir-4475	2	76	,	,	PUNCT
asir-4475	2	77	also	also	ADV
asir-4475	2	78	known	know	VERB
asir-4475	2	79	as	as	ADP
asir-4475	2	80	the	the	DET
asir-4475	2	81	“	"	PUNCT
asir-4475	2	82	riemann	riemann	PROPN
asir-4475	2	83	’s	’s	PART
asir-4475	2	84	hypothesis	hypothesis	NOUN
asir-4475	2	85	”	"	PUNCT
asir-4475	2	86	,	,	PUNCT
asir-4475	2	87	rh	rh	PROPN
asir-4475	2	88	)	)	PUNCT
asir-4475	2	89	,	,	PUNCT
asir-4475	2	90	now	now	ADV
asir-4475	2	91	the	the	DET
asir-4475	2	92	author	author	NOUN
asir-4475	2	93	provides	provide	VERB
asir-4475	2	94	a	a	DET
asir-4475	2	95	new	new	ADJ
asir-4475	2	96	proof	proof	NOUN
asir-4475	2	97	,	,	PUNCT
asir-4475	2	98	using	use	VERB
asir-4475	2	99	the	the	DET
asir-4475	2	100	“	"	PUNCT
asir-4475	2	101	spira	spira	PROPN
asir-4475	2	102	criterion	criterion	NOUN
asir-4475	2	103	”	"	PUNCT
asir-4475	2	104	that	that	PRON
asir-4475	2	105	states	state	VERB
asir-4475	2	106	“	"	PUNCT
asir-4475	2	107	the	the	DET
asir-4475	2	108	rh	rh	PROPN
asir-4475	2	109	is	be	AUX
asir-4475	2	110	equivalent	equivalent	ADJ
asir-4475	2	111	to	to	ADP
asir-4475	2	112	the	the	DET
asir-4475	2	113	statement	statement	NOUN
asir-4475	2	114	that	that	SCONJ
asir-4475	2	115	if	if	SCONJ
asir-4475	2	116	>0.5	>0.5	NUM
asir-4475	2	117	and	and	CCONJ
asir-4475	2	118	t	t	X
asir-4475	2	119	>	>	X
asir-4475	2	120	6.5	6.5	NUM
asir-4475	2	121	then	then	ADV
asir-4475	2	122	|(1	|(1	NOUN
asir-4475	2	123	-	-	PUNCT
asir-4475	2	124	s)|	s)|	PROPN
asir-4475	2	125	>	>	X
asir-4475	2	126	|(s)|	|(s)|	PROPN
asir-4475	2	127	”	"	PUNCT
asir-4475	2	128	.	.	PUNCT
asir-4475	3	1	we	we	PRON
asir-4475	3	2	use	use	VERB
asir-4475	3	3	the	the	DET
asir-4475	3	4	concept	concept	NOUN
asir-4475	3	5	of	of	ADP
asir-4475	3	6	“	"	PUNCT
asir-4475	3	7	transfer	transfer	NOUN
asir-4475	3	8	function	function	NOUN
asir-4475	3	9	”	"	PUNCT
asir-4475	3	10	for	for	ADP
asir-4475	3	11	control	control	NOUN
asir-4475	3	12	systems	system	NOUN
asir-4475	3	13	.	.	PUNCT
asir-4475	4	1	this	this	DET
asir-4475	4	2	new	new	ADJ
asir-4475	4	3	proof	proof	NOUN
asir-4475	4	4	is	be	AUX
asir-4475	4	5	so	so	ADV
asir-4475	4	6	simple	simple	ADJ
asir-4475	4	7	that	that	SCONJ
asir-4475	4	8	the	the	DET
asir-4475	4	9	author	author	NOUN
asir-4475	4	10	wonders	wonder	VERB
asir-4475	4	11	why	why	SCONJ
asir-4475	4	12	a	a	DET
asir-4475	4	13	great	great	ADJ
asir-4475	4	14	mathematician	mathematician	NOUN
asir-4475	4	15	like	like	ADP
asir-4475	4	16	riemann	riemann	PROPN
asir-4475	4	17	did	do	AUX
asir-4475	4	18	not	not	PART
asir-4475	4	19	see	see	VERB
asir-4475	4	20	it	it	PRON
asir-4475	4	21	;	;	PUNCT
asir-4475	4	22	therefore	therefore	ADV
asir-4475	4	23	f.	f.	PROPN
asir-4475	4	24	galetto	galetto	PROPN
asir-4475	4	25	thinks	think	VERB
asir-4475	4	26	that	that	SCONJ
asir-4475	4	27	somewhere	somewhere	ADV
asir-4475	4	28	in	in	ADP
asir-4475	4	29	the	the	DET
asir-4475	4	30	purported	purport	VERB
asir-4475	4	31	proof	proof	NOUN
asir-4475	4	32	there	there	PRON
asir-4475	4	33	should	should	AUX
asir-4475	4	34	be	be	AUX
asir-4475	4	35	an	an	DET
asir-4475	4	36	error	error	NOUN
asir-4475	4	37	.	.	PUNCT
asir-4475	5	1	keywords	keyword	NOUN
asir-4475	5	2	zeta	zeta	PROPN
asir-4475	5	3	function	function	PROPN
asir-4475	5	4	,	,	PUNCT
asir-4475	5	5	functional	functional	ADJ
asir-4475	5	6	equation	equation	NOUN
asir-4475	5	7	,	,	PUNCT
asir-4475	5	8	nontrivial	nontrivial	NOUN
asir-4475	5	9	zeros	zero	NOUN
asir-4475	5	10	,	,	PUNCT
asir-4475	5	11	critical	critical	ADJ
asir-4475	5	12	line	line	NOUN
asir-4475	5	13	,	,	PUNCT
asir-4475	5	14	spira	spira	PROPN
asir-4475	5	15	criterion	criterion	NOUN
asir-4475	5	16	,	,	PUNCT
asir-4475	5	17	generalised	generalise	VERB
asir-4475	5	18	power	power	NOUN
asir-4475	5	19	transfer	transfer	NOUN
asir-4475	5	20	function	function	NOUN
asir-4475	5	21	.	.	PUNCT
asir-4475	6	1	1	1	X
asir-4475	6	2	.	.	X
asir-4475	6	3	introduction	introduction	NOUN
asir-4475	6	4	it	it	PRON
asir-4475	6	5	is	be	AUX
asir-4475	6	6	well	well	ADV
asir-4475	6	7	known	know	VERB
asir-4475	6	8	that	that	SCONJ
asir-4475	6	9	for	for	ADP
asir-4475	6	10	over	over	ADP
asir-4475	6	11	a	a	DET
asir-4475	6	12	century	century	NOUN
asir-4475	6	13	mathematicians	mathematician	NOUN
asir-4475	6	14	have	have	AUX
asir-4475	6	15	been	be	AUX
asir-4475	6	16	trying	try	VERB
asir-4475	6	17	to	to	PART
asir-4475	6	18	prove	prove	VERB
asir-4475	6	19	the	the	DET
asir-4475	6	20	so	so	ADV
asir-4475	6	21	-	-	PUNCT
asir-4475	6	22	called	call	VERB
asir-4475	6	23	riemann	riemann	PROPN
asir-4475	6	24	hypothesis	hypothesis	NOUN
asir-4475	6	25	,	,	PUNCT
asir-4475	6	26	rh	rh	PROPN
asir-4475	6	27	for	for	ADP
asir-4475	6	28	short	short	ADJ
asir-4475	6	29	,	,	PUNCT
asir-4475	6	30	a	a	DET
asir-4475	6	31	conjecture	conjecture	NOUN
asir-4475	6	32	claimed	claim	VERB
asir-4475	6	33	by	by	ADP
asir-4475	6	34	riemann	riemann	PROPN
asir-4475	7	1	[	[	X
asir-4475	7	2	who	who	PRON
asir-4475	7	3	was	be	AUX
asir-4475	7	4	professor	professor	NOUN
asir-4475	7	5	at	at	ADP
asir-4475	7	6	university	university	PROPN
asir-4475	7	7	of	of	ADP
asir-4475	7	8	gottingen	gottingen	NOUN
asir-4475	7	9	in	in	ADP
asir-4475	7	10	germany	germany	PROPN
asir-4475	7	11	]	]	PUNCT
asir-4475	7	12	,	,	PUNCT
asir-4475	7	13	near	near	ADP
asir-4475	7	14	1859	1859	NUM
asir-4475	7	15	in	in	ADP
asir-4475	7	16	a	a	DET
asir-4475	7	17	8	8	NUM
asir-4475	7	18	-	-	PUNCT
asir-4475	7	19	page	page	NOUN
asir-4475	7	20	paper	paper	NOUN
asir-4475	7	21	“	"	PUNCT
asir-4475	7	22	on	on	ADP
asir-4475	7	23	the	the	DET
asir-4475	7	24	number	number	NOUN
asir-4475	7	25	of	of	ADP
asir-4475	7	26	primes	prime	NOUN
asir-4475	7	27	less	less	ADJ
asir-4475	7	28	than	than	ADP
asir-4475	7	29	a	a	DET
asir-4475	7	30	given	give	VERB
asir-4475	7	31	magnitude	magnitude	NOUN
asir-4475	7	32	”	"	PUNCT
asir-4475	7	33	shown	show	VERB
asir-4475	7	34	at	at	ADP
asir-4475	7	35	berlin	berlin	PROPN
asir-4475	7	36	academy	academy	PROPN
asir-4475	7	37	,	,	PUNCT
asir-4475	7	38	and	and	CCONJ
asir-4475	7	39	dated	date	VERB
asir-4475	7	40	/	/	SYM
asir-4475	7	41	published	publish	VERB
asir-4475	7	42	in	in	ADP
asir-4475	7	43	1859	1859	NUM
asir-4475	7	44	;	;	PUNCT
asir-4475	7	45	it	it	PRON
asir-4475	7	46	is	be	AUX
asir-4475	7	47	well	well	ADV
asir-4475	7	48	known	know	VERB
asir-4475	7	49	,	,	PUNCT
asir-4475	7	50	as	as	ADV
asir-4475	7	51	well	well	ADV
asir-4475	7	52	,	,	PUNCT
asir-4475	7	53	that	that	SCONJ
asir-4475	7	54	rh	rh	PROPN
asir-4475	7	55	is	be	AUX
asir-4475	7	56	related	relate	VERB
asir-4475	7	57	to	to	PART
asir-4475	7	58	set	set	VERB
asir-4475	7	59	of	of	ADP
asir-4475	7	60	all	all	DET
asir-4475	7	61	the	the	DET
asir-4475	7	62	prime	prime	ADJ
asir-4475	7	63	numbers	number	NOUN
asir-4475	7	64	(	(	PUNCT
asir-4475	7	65	figure	figure	NOUN
asir-4475	7	66	1	1	NUM
asir-4475	7	67	)	)	PUNCT
asir-4475	7	68	;	;	PUNCT
asir-4475	7	69	prime	prime	ADJ
asir-4475	7	70	numbers	number	NOUN
asir-4475	7	71	are	be	AUX
asir-4475	7	72	fundamental	fundamental	ADJ
asir-4475	7	73	for	for	ADP
asir-4475	7	74	encryption	encryption	NOUN
asir-4475	7	75	of	of	ADP
asir-4475	7	76	documents	document	NOUN
asir-4475	7	77	and	and	CCONJ
asir-4475	7	78	data	datum	NOUN
asir-4475	7	79	(	(	PUNCT
asir-4475	7	80	e.g.	e.g.	ADV
asir-4475	7	81	payments	payment	NOUN
asir-4475	7	82	with	with	ADP
asir-4475	7	83	credit	credit	NOUN
asir-4475	7	84	cards	card	NOUN
asir-4475	7	85	)	)	PUNCT
asir-4475	7	86	.	.	PUNCT
asir-4475	8	1	he	he	PRON
asir-4475	8	2	invented	invent	VERB
asir-4475	8	3	the	the	DET
asir-4475	8	4	riemann	riemann	PROPN
asir-4475	8	5	zeta	zeta	PROPN
asir-4475	8	6	function	function	PROPN
asir-4475	8	7	:	:	PUNCT
asir-4475	8	8	(z	(z	NOUN
asir-4475	8	9	)	)	PUNCT
asir-4475	8	10	,	,	PUNCT
asir-4475	8	11	where	where	SCONJ
asir-4475	8	12	z	z	NOUN
asir-4475	8	13	is	be	AUX
asir-4475	8	14	a	a	DET
asir-4475	8	15	complex	complex	ADJ
asir-4475	8	16	number	number	NOUN
asir-4475	8	17	z	z	NOUN
asir-4475	8	18	=	=	NOUN
asir-4475	8	19	x+iy	x+iy	NOUN
asir-4475	9	1	and	and	CCONJ
asir-4475	9	2	i	i	PRON
asir-4475	9	3	is	be	AUX
asir-4475	9	4	the	the	DET
asir-4475	9	5	“	"	PUNCT
asir-4475	9	6	positive	positive	ADJ
asir-4475	9	7	”	"	PUNCT
asir-4475	9	8	imaginary	imaginary	ADJ
asir-4475	9	9	unit	unit	NOUN
asir-4475	9	10	such	such	ADJ
asir-4475	9	11	that	that	DET
asir-4475	9	12	i2=-1	i2=-1	PROPN
asir-4475	9	13	.	.	PUNCT
asir-4475	10	1	in	in	ADP
asir-4475	10	2	the	the	DET
asir-4475	10	3	following	follow	VERB
asir-4475	10	4	part	part	NOUN
asir-4475	10	5	of	of	ADP
asir-4475	10	6	the	the	DET
asir-4475	10	7	paper	paper	NOUN
asir-4475	10	8	we	we	PRON
asir-4475	10	9	will	will	AUX
asir-4475	10	10	use	use	VERB
asir-4475	10	11	the	the	DET
asir-4475	10	12	symbols	symbol	NOUN
asir-4475	10	13	s=+it	s=+it	NOUN
asir-4475	10	14	for	for	ADP
asir-4475	10	15	the	the	DET
asir-4475	10	16	complex	complex	ADJ
asir-4475	10	17	variable	variable	NOUN
asir-4475	10	18	,	,	PUNCT
asir-4475	10	19	as	as	ADP
asir-4475	10	20	in	in	ADP
asir-4475	10	21	titchmarsh	titchmarsh	NOUN
asir-4475	10	22	(	(	PUNCT
asir-4475	10	23	1986	1986	NUM
asir-4475	10	24	)	)	PUNCT
asir-4475	10	25	.	.	PUNCT
asir-4475	11	1	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	PROPN
asir-4475	11	2	applied	apply	VERB
asir-4475	11	3	science	science	NOUN
asir-4475	11	4	and	and	CCONJ
asir-4475	11	5	innovative	innovative	ADJ
asir-4475	11	6	research	research	NOUN
asir-4475	11	7	vol	vol	NOUN
asir-4475	11	8	.	.	PROPN
asir-4475	12	1	6	6	NUM
asir-4475	12	2	,	,	PUNCT
asir-4475	12	3	no	no	INTJ
asir-4475	12	4	.	.	NOUN
asir-4475	12	5	1	1	NUM
asir-4475	12	6	,	,	PUNCT
asir-4475	12	7	2022	2022	NUM
asir-4475	12	8	15	15	NUM
asir-4475	12	9	published	publish	VERB
asir-4475	12	10	by	by	ADP
asir-4475	12	11	scholink	scholink	PROPN
asir-4475	12	12	inc	inc	PROPN
asir-4475	12	13	.	.	PROPN
asir-4475	12	14	…	…	PUNCT
asir-4475	12	15	…	…	PUNCT
asir-4475	12	16	…	…	SYM
asir-4475	12	17	……	……	NOUN
asir-4475	12	18	figure	figure	NOUN
asir-4475	12	19	1	1	NUM
asir-4475	12	20	.	.	PUNCT
asir-4475	13	1	from	from	ADP
asir-4475	13	2	the	the	DET
asir-4475	13	3	original	original	ADJ
asir-4475	13	4	b.	b.	PROPN
asir-4475	13	5	riemann	riemann	PROPN
asir-4475	13	6	manuscript	manuscript	PROPN
asir-4475	13	7	the	the	DET
asir-4475	13	8	description	description	NOUN
asir-4475	13	9	the	the	DET
asir-4475	13	10	zeta	zeta	PROPN
asir-4475	13	11	function	function	NOUN
asir-4475	13	12	(𝑠	(𝑠	NOUN
asir-4475	13	13	)	)	PUNCT
asir-4475	14	1	=	=	SYM
asir-4475	15	1	(𝜎	(𝜎	PROPN
asir-4475	15	2	+	+	CCONJ
asir-4475	15	3	𝑖𝑡	𝑖𝑡	NOUN
asir-4475	15	4	)	)	PUNCT
asir-4475	15	5	will	will	AUX
asir-4475	15	6	not	not	PART
asir-4475	15	7	be	be	AUX
asir-4475	15	8	introduced	introduce	VERB
asir-4475	15	9	here	here	ADV
asir-4475	15	10	,	,	PUNCT
asir-4475	15	11	interested	interested	ADJ
asir-4475	15	12	readers	reader	NOUN
asir-4475	15	13	can	can	AUX
asir-4475	15	14	refer	refer	VERB
asir-4475	15	15	to	to	ADP
asir-4475	15	16	any	any	DET
asir-4475	15	17	text	text	NOUN
asir-4475	15	18	book	book	NOUN
asir-4475	15	19	,	,	PUNCT
asir-4475	15	20	e.g.	e.g.	ADV
asir-4475	15	21	,	,	PUNCT
asir-4475	15	22	titchmarsh	titchmarsh	NOUN
asir-4475	15	23	(	(	PUNCT
asir-4475	15	24	1986	1986	NUM
asir-4475	15	25	)	)	PUNCT
asir-4475	15	26	,	,	PUNCT
asir-4475	15	27	bombieri	bombieri	NOUN
asir-4475	15	28	(	(	PUNCT
asir-4475	15	29	2000	2000	NUM
asir-4475	15	30	)	)	PUNCT
asir-4475	15	31	,	,	PUNCT
asir-4475	15	32	broughan	broughan	ADP
asir-4475	15	33	(	(	PUNCT
asir-4475	15	34	2017	2017	NUM
asir-4475	15	35	)	)	PUNCT
asir-4475	15	36	.	.	PUNCT
asir-4475	16	1	(𝑠	(𝑠	NOUN
asir-4475	16	2	)	)	PUNCT
asir-4475	16	3	is	be	AUX
asir-4475	16	4	a	a	DET
asir-4475	16	5	meromorphic	meromorphic	ADJ
asir-4475	16	6	function	function	NOUN
asir-4475	16	7	with	with	ADP
asir-4475	16	8	only	only	ADV
asir-4475	16	9	a	a	DET
asir-4475	16	10	simple	simple	ADJ
asir-4475	16	11	pole	pole	NOUN
asir-4475	16	12	at	at	ADP
asir-4475	16	13	s=1	s=1	X
asir-4475	17	1	[	[	X
asir-4475	17	2	with	with	ADP
asir-4475	17	3	residue	residue	NOUN
asir-4475	17	4	1	1	NUM
asir-4475	17	5	]	]	PUNCT
asir-4475	17	6	;	;	PUNCT
asir-4475	17	7	moreover	moreover	ADV
asir-4475	17	8	(𝑠)0	(𝑠)0	AUX
asir-4475	17	9	for	for	ADP
asir-4475	17	10	all	all	DET
asir-4475	17	11	z	z	NOUN
asir-4475	17	12			PROPN
asir-4475	17	13	c	c	NOUN
asir-4475	17	14	with	with	ADP
asir-4475	17	15	re(s)=>1	re(s)=>1	NOUN
asir-4475	17	16	;	;	PUNCT
asir-4475	17	17	it	it	PRON
asir-4475	17	18	has	have	VERB
asir-4475	17	19	zeros	zero	NOUN
asir-4475	17	20	at	at	ADP
asir-4475	17	21	the	the	DET
asir-4475	17	22	negative	negative	ADJ
asir-4475	17	23	even	even	ADV
asir-4475	17	24	integers	integer	NOUN
asir-4475	17	25	-2	-2	INTJ
asir-4475	17	26	,	,	PUNCT
asir-4475	17	27	-4	-4	INTJ
asir-4475	17	28	,	,	PUNCT
asir-4475	17	29	-6	-6	ADV
asir-4475	17	30	,	,	PUNCT
asir-4475	17	31	…	…	PUNCT
asir-4475	17	32	..	..	PUNCT
asir-4475	17	33	,	,	PUNCT
asir-4475	17	34	named	name	VERB
asir-4475	17	35	trivial	trivial	ADJ
asir-4475	17	36	zeros	zero	NOUN
asir-4475	17	37	.	.	PUNCT
asir-4475	18	1	the	the	DET
asir-4475	18	2	other	other	ADJ
asir-4475	18	3	zeros	zero	NOUN
asir-4475	18	4	are	be	AUX
asir-4475	18	5	named	name	VERB
asir-4475	18	6	nontrivial	nontrivial	ADJ
asir-4475	18	7	zeros	zero	NOUN
asir-4475	18	8	:	:	PUNCT
asir-4475	18	9	all	all	DET
asir-4475	18	10	the	the	DET
asir-4475	18	11	“	"	PUNCT
asir-4475	18	12	known	know	VERB
asir-4475	18	13	”	"	PUNCT
asir-4475	18	14	zeros	zero	NOUN
asir-4475	18	15	,	,	PUNCT
asir-4475	18	16	computed	compute	VERB
asir-4475	18	17	up	up	ADP
asir-4475	18	18	to	to	ADP
asir-4475	18	19	now	now	ADV
asir-4475	18	20	[	[	X
asir-4475	18	21	up	up	ADP
asir-4475	18	22	to	to	ADP
asir-4475	18	23	2004	2004	NUM
asir-4475	18	24	,	,	PUNCT
asir-4475	18	25	1012	1012	NUM
asir-4475	18	26	zeros	zero	NOUN
asir-4475	18	27	have	have	AUX
asir-4475	18	28	been	be	AUX
asir-4475	18	29	computed	compute	VERB
asir-4475	18	30	,	,	PUNCT
asir-4475	18	31	all	all	ADV
asir-4475	18	32	on	on	ADP
asir-4475	18	33	the	the	DET
asir-4475	18	34	critical	critical	ADJ
asir-4475	18	35	line	line	NOUN
asir-4475	18	36	]	]	PUNCT
asir-4475	18	37	,	,	PUNCT
asir-4475	18	38	are	be	AUX
asir-4475	18	39	the	the	DET
asir-4475	18	40	complex	complex	ADJ
asir-4475	18	41	numbers	number	NOUN
asir-4475	18	42	s=1/2	s=1/2	VERB
asir-4475	18	43	+	+	CCONJ
asir-4475	18	44	it	it	PRON
asir-4475	18	45	,	,	PUNCT
asir-4475	18	46	with	with	ADP
asir-4475	18	47	suitable	suitable	ADJ
asir-4475	18	48	values	value	NOUN
asir-4475	18	49	of	of	ADP
asir-4475	18	50	t.	t.	NOUN
asir-4475	18	51	it	it	PRON
asir-4475	18	52	satisfies	satisfy	VERB
asir-4475	18	53	the	the	DET
asir-4475	18	54	functional	functional	ADJ
asir-4475	18	55	equation	equation	NOUN
asir-4475	18	56	π−	π−	NOUN
asir-4475	18	57	𝑠	𝑠	PROPN
asir-4475	18	58	2	2	NUM
asir-4475	18	59	γ	γ	X
asir-4475	18	60	(	(	PUNCT
asir-4475	18	61	𝑠	𝑠	PROPN
asir-4475	18	62	2	2	NUM
asir-4475	18	63	)	)	PUNCT
asir-4475	18	64	(s	(s	NOUN
asir-4475	18	65	)	)	PUNCT
asir-4475	18	66	=	=	PUNCT
asir-4475	19	1	π−(1−𝑠)/2	π−(1−𝑠)/2	VERB
asir-4475	19	2	γ[(1	γ[(1	PROPN
asir-4475	19	3	−	−	PROPN
asir-4475	19	4	𝑠)/2])(1	𝑠)/2])(1	PROPN
asir-4475	19	5	−	−	PROPN
asir-4475	19	6	s	s	NOUN
asir-4475	19	7	)	)	PUNCT
asir-4475	19	8	(	(	PUNCT
asir-4475	19	9	1	1	X
asir-4475	19	10	)	)	PUNCT
asir-4475	19	11	the	the	DET
asir-4475	19	12	properties	property	NOUN
asir-4475	19	13	of	of	ADP
asir-4475	19	14	(s	(s	NOUN
asir-4475	19	15	)	)	PUNCT
asir-4475	19	16	are	be	AUX
asir-4475	19	17	as	as	SCONJ
asir-4475	19	18	follows	follow	VERB
asir-4475	19	19	:	:	PUNCT
asir-4475	19	20	a	a	X
asir-4475	19	21	)	)	PUNCT
asir-4475	19	22	(s	(s	NOUN
asir-4475	19	23	)	)	PUNCT
asir-4475	19	24	has	have	VERB
asir-4475	19	25	no	no	DET
asir-4475	19	26	zero	zero	NUM
asir-4475	19	27	for	for	ADP
asir-4475	19	28	re(s)>1	re(s)>1	NOUN
asir-4475	19	29	;	;	PUNCT
asir-4475	19	30	b	b	X
asir-4475	19	31	)	)	PUNCT
asir-4475	19	32	the	the	DET
asir-4475	19	33	only	only	ADJ
asir-4475	19	34	pole	pole	NOUN
asir-4475	19	35	(s	(s	NOUN
asir-4475	19	36	)	)	PUNCT
asir-4475	19	37	is	be	AUX
asir-4475	19	38	at	at	ADP
asir-4475	19	39	s=1	s=1	ADP
asir-4475	19	40	:	:	PUNCT
asir-4475	19	41	it	it	PRON
asir-4475	19	42	is	be	AUX
asir-4475	19	43	simple	simple	ADJ
asir-4475	19	44	and	and	CCONJ
asir-4475	19	45	has	have	VERB
asir-4475	19	46	residue	residue	NOUN
asir-4475	19	47	1	1	NUM
asir-4475	19	48	;	;	PUNCT
asir-4475	19	49	c	c	X
asir-4475	19	50	)	)	PUNCT
asir-4475	20	1	(s	(s	NOUN
asir-4475	20	2	)	)	PUNCT
asir-4475	21	1	has	have	VERB
asir-4475	21	2	trivial	trivial	ADJ
asir-4475	21	3	zeros	zero	NOUN
asir-4475	21	4	at	at	ADP
asir-4475	21	5	the	the	DET
asir-4475	21	6	negative	negative	ADJ
asir-4475	21	7	even	even	ADV
asir-4475	21	8	integers	integer	NOUN
asir-4475	21	9	z	z	NOUN
asir-4475	21	10	=	=	SYM
asir-4475	21	11	-2	-2	INTJ
asir-4475	21	12	,	,	PUNCT
asir-4475	21	13	-4	-4	INTJ
asir-4475	21	14	,	,	PUNCT
asir-4475	21	15	-6	-6	ADV
asir-4475	21	16	,	,	PUNCT
asir-4475	21	17	…	…	PUNCT
asir-4475	21	18	.	.	PUNCT
asir-4475	22	1	d	d	X
asir-4475	22	2	)	)	PUNCT
asir-4475	22	3	all	all	DET
asir-4475	22	4	the	the	DET
asir-4475	22	5	nontrivial	nontrivial	ADJ
asir-4475	22	6	zeros	zero	NOUN
asir-4475	22	7	lie	lie	VERB
asir-4475	22	8	inside	inside	ADP
asir-4475	22	9	the	the	DET
asir-4475	22	10	region	region	NOUN
asir-4475	22	11	,	,	PUNCT
asir-4475	22	12	named	name	VERB
asir-4475	22	13	critical	critical	ADJ
asir-4475	22	14	strip	strip	NOUN
asir-4475	22	15	,	,	PUNCT
asir-4475	22	16	0	0	NUM
asir-4475	22	17			NOUN
asir-4475	22	18	re(s	re(s	ADJ
asir-4475	22	19	)	)	PUNCT
asir-4475	22	20			NOUN
asir-4475	22	21	1	1	NUM
asir-4475	22	22	and	and	CCONJ
asir-4475	22	23	are	be	AUX
asir-4475	22	24	symmetric	symmetric	ADJ
asir-4475	22	25	about	about	ADP
asir-4475	22	26	both	both	CCONJ
asir-4475	22	27	the	the	DET
asir-4475	22	28	vertical	vertical	ADJ
asir-4475	22	29	line	line	NOUN
asir-4475	22	30	,	,	PUNCT
asir-4475	22	31	named	name	VERB
asir-4475	22	32	critical	critical	ADJ
asir-4475	22	33	line	line	NOUN
asir-4475	22	34	,	,	PUNCT
asir-4475	22	35	re(z)==1/2	re(z)==1/2	PROPN
asir-4475	22	36	and	and	CCONJ
asir-4475	22	37	the	the	DET
asir-4475	22	38	real	real	ADJ
asir-4475	22	39	axis	axis	NOUN
asir-4475	22	40	im(z)=0	im(z)=0	PROPN
asir-4475	22	41	:	:	PUNCT
asir-4475	22	42	(𝑠	(𝑠	NOUN
asir-4475	22	43	)	)	PUNCT
asir-4475	23	1	=	=	SYM
asir-4475	23	2	(𝑠)̅̅	(𝑠)̅̅	PROPN
asir-4475	23	3	̅̅	̅̅	PROPN
asir-4475	23	4	̅̅	̅̅	PROPN
asir-4475	23	5	=	=	PROPN
asir-4475	24	1	(	(	PROPN
asir-4475	24	2	�	�	PROPN
asir-4475	24	3	̅	̅	NOUN
asir-4475	24	4	�	�	NOUN
asir-4475	24	5	)	)	PUNCT
asir-4475	24	6	=	=	PUNCT
asir-4475	25	1	(1	(1	VERB
asir-4475	25	2	−	−	NOUN
asir-4475	25	3	𝑠	𝑠	NOUN
asir-4475	25	4	)	)	PUNCT
asir-4475	25	5	=	=	PUNCT
asir-4475	26	1	(1	(1	NOUN
asir-4475	26	2	−	−	NOUN
asir-4475	26	3	�	�	NOUN
asir-4475	26	4	̅	̅	NOUN
asir-4475	26	5	�	�	NOUN
asir-4475	26	6	)	)	PUNCT
asir-4475	26	7	riemann	riemann	PROPN
asir-4475	26	8	conjectured	conjecture	VERB
asir-4475	26	9	the	the	DET
asir-4475	26	10	so	so	ADV
asir-4475	26	11	-	-	PUNCT
asir-4475	26	12	called	call	VERB
asir-4475	26	13	riemann	riemann	PROPN
asir-4475	26	14	hypothesis	hypothesis	NOUN
asir-4475	26	15	(	(	PUNCT
asir-4475	26	16	rh	rh	PROPN
asir-4475	26	17	)	)	PUNCT
asir-4475	26	18	,	,	PUNCT
asir-4475	26	19	or	or	CCONJ
asir-4475	26	20	conjecture	conjecture	NOUN
asir-4475	26	21	(	(	PUNCT
asir-4475	26	22	rc	rc	PROPN
asir-4475	26	23	):	):	PUNCT
asir-4475	26	24	rc	rc	PROPN
asir-4475	26	25	states	state	VERB
asir-4475	26	26	all	all	DET
asir-4475	26	27	nontrivial	nontrivial	ADJ
asir-4475	26	28	zeros	zero	NOUN
asir-4475	26	29	of	of	ADP
asir-4475	26	30	(s	(s	NOUN
asir-4475	26	31	)	)	PUNCT
asir-4475	26	32	have	have	VERB
asir-4475	26	33	real	real	ADJ
asir-4475	26	34	part	part	NOUN
asir-4475	26	35			PUNCT
asir-4475	26	36	equal	equal	ADJ
asir-4475	26	37	to	to	ADP
asir-4475	26	38	1/2	1/2	NUM
asir-4475	26	39	.	.	PUNCT
asir-4475	27	1	many	many	ADJ
asir-4475	27	2	great	great	ADJ
asir-4475	27	3	mathematicians	mathematician	NOUN
asir-4475	27	4	tackled	tackle	VERB
asir-4475	27	5	this	this	DET
asir-4475	27	6	problem	problem	NOUN
asir-4475	27	7	;	;	PUNCT
asir-4475	27	8	we	we	PRON
asir-4475	27	9	do	do	AUX
asir-4475	27	10	not	not	PART
asir-4475	27	11	mention	mention	VERB
asir-4475	27	12	them	they	PRON
asir-4475	27	13	,	,	PUNCT
asir-4475	27	14	because	because	SCONJ
asir-4475	27	15	they	they	PRON
asir-4475	27	16	can	can	AUX
asir-4475	27	17	be	be	AUX
asir-4475	27	18	found	find	VERB
asir-4475	27	19	in	in	ADP
asir-4475	27	20	many	many	ADJ
asir-4475	27	21	books	book	NOUN
asir-4475	27	22	and	and	CCONJ
asir-4475	27	23	papers	paper	NOUN
asir-4475	27	24	.	.	PUNCT
asir-4475	28	1	if	if	SCONJ
asir-4475	28	2	rc	rc	PROPN
asir-4475	28	3	would	would	AUX
asir-4475	28	4	be	be	AUX
asir-4475	28	5	related	relate	VERB
asir-4475	28	6	to	to	ADP
asir-4475	28	7	physics	physics	NOUN
asir-4475	28	8	,	,	PUNCT
asir-4475	28	9	it	it	PRON
asir-4475	28	10	would	would	AUX
asir-4475	28	11	be	be	AUX
asir-4475	28	12	considered	consider	VERB
asir-4475	28	13	a	a	DET
asir-4475	28	14	“	"	PUNCT
asir-4475	28	15	universal	universal	ADJ
asir-4475	28	16	law	law	NOUN
asir-4475	28	17	”	"	PUNCT
asir-4475	28	18	:	:	PUNCT
asir-4475	28	19	up	up	ADP
asir-4475	28	20	to	to	ADP
asir-4475	28	21	2004	2004	NUM
asir-4475	28	22	,	,	PUNCT
asir-4475	28	23	1012	1012	NUM
asir-4475	28	24	zeros	zero	NOUN
asir-4475	28	25	have	have	AUX
asir-4475	28	26	been	be	AUX
asir-4475	28	27	computed	compute	VERB
asir-4475	28	28	,	,	PUNCT
asir-4475	28	29	all	all	ADV
asir-4475	28	30	on	on	ADP
asir-4475	28	31	the	the	DET
asir-4475	28	32	critical	critical	ADJ
asir-4475	28	33	line	line	NOUN
asir-4475	28	34	.	.	PUNCT
asir-4475	29	1	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	PROPN
asir-4475	29	2	applied	apply	VERB
asir-4475	29	3	science	science	NOUN
asir-4475	29	4	and	and	CCONJ
asir-4475	29	5	innovative	innovative	ADJ
asir-4475	29	6	research	research	NOUN
asir-4475	29	7	vol	vol	NOUN
asir-4475	29	8	.	.	PROPN
asir-4475	30	1	6	6	NUM
asir-4475	30	2	,	,	PUNCT
asir-4475	30	3	no	no	INTJ
asir-4475	30	4	.	.	NOUN
asir-4475	30	5	1	1	NUM
asir-4475	30	6	,	,	PUNCT
asir-4475	30	7	2022	2022	NUM
asir-4475	30	8	16	16	NUM
asir-4475	30	9	published	publish	VERB
asir-4475	30	10	by	by	ADP
asir-4475	30	11	scholink	scholink	PROPN
asir-4475	30	12	inc	inc	PROPN
asir-4475	30	13	.	.	PUNCT
asir-4475	31	1	if	if	SCONJ
asir-4475	31	2	rc	rc	PROPN
asir-4475	31	3	would	would	AUX
asir-4475	31	4	be	be	AUX
asir-4475	31	5	related	relate	VERB
asir-4475	31	6	to	to	ADP
asir-4475	31	7	statistics	statistic	NOUN
asir-4475	31	8	,	,	PUNCT
asir-4475	31	9	<	<	X
asir-4475	31	10	<	<	X
asir-4475	31	11	the	the	DET
asir-4475	31	12	hypothesis	hypothesis	NOUN
asir-4475	31	13	h0	h0	NOUN
asir-4475	31	14	:	:	PUNCT
asir-4475	31	15	“	"	PUNCT
asir-4475	31	16	the	the	DET
asir-4475	31	17	nontrivial	nontrivial	ADJ
asir-4475	31	18	zeros	zero	NOUN
asir-4475	31	19	are	be	AUX
asir-4475	31	20	on	on	ADP
asir-4475	31	21	the	the	DET
asir-4475	31	22	critical	critical	ADJ
asir-4475	31	23	line	line	NOUN
asir-4475	31	24	”	"	PUNCT
asir-4475	31	25	>	>	X
asir-4475	31	26	>	>	X
asir-4475	31	27	,	,	PUNCT
asir-4475	31	28	would	would	AUX
asir-4475	31	29	be	be	AUX
asir-4475	31	30	confirmed	confirm	VERB
asir-4475	31	31	with	with	ADP
asir-4475	31	32	a	a	DET
asir-4475	31	33	confidence	confidence	NOUN
asir-4475	31	34	level	level	NOUN
asir-4475	31	35	(	(	PUNCT
asir-4475	31	36	cl	cl	NOUN
asir-4475	31	37	)	)	PUNCT
asir-4475	31	38	>	>	X
asir-4475	32	1	0.9999999999	0.9999999999	NUM
asir-4475	32	2	:	:	PUNCT
asir-4475	32	3	the	the	DET
asir-4475	32	4	evidence	evidence	NOUN
asir-4475	32	5	of	of	ADP
asir-4475	32	6	1012	1012	NUM
asir-4475	32	7	zeros	zero	NOUN
asir-4475	32	8	computed	compute	VERB
asir-4475	32	9	,	,	PUNCT
asir-4475	32	10	all	all	ADV
asir-4475	32	11	on	on	ADP
asir-4475	32	12	the	the	DET
asir-4475	32	13	critical	critical	ADJ
asir-4475	32	14	line	line	NOUN
asir-4475	32	15	(	(	PUNCT
asir-4475	32	16	as	as	ADP
asir-4475	32	17	to	to	ADP
asir-4475	32	18	2004	2004	NUM
asir-4475	32	19	)	)	PUNCT
asir-4475	32	20	supports	support	VERB
asir-4475	32	21	h0	h0	NOUN
asir-4475	32	22	with	with	ADP
asir-4475	32	23	that	that	DET
asir-4475	32	24	“	"	PUNCT
asir-4475	32	25	high	high	ADJ
asir-4475	32	26	”	"	PUNCT
asir-4475	32	27	cl	cl	NOUN
asir-4475	32	28	.	.	PUNCT
asir-4475	33	1	but	but	CCONJ
asir-4475	33	2	mathematics	mathematics	PROPN
asir-4475	33	3	asks	ask	VERB
asir-4475	33	4	much	much	ADV
asir-4475	33	5	more	more	ADJ
asir-4475	33	6	than	than	ADP
asir-4475	33	7	statistics	statistic	NOUN
asir-4475	33	8	and	and	CCONJ
asir-4475	33	9	physics	physic	NOUN
asir-4475	33	10	…	…	PUNCT
asir-4475	33	11	also	also	ADV
asir-4475	33	12	a	a	DET
asir-4475	33	13	theorem	theorem	NOUN
asir-4475	33	14	of	of	ADP
asir-4475	33	15	g.	g.	PROPN
asir-4475	33	16	hardy	hardy	PROPN
asir-4475	34	1	[	[	X
asir-4475	34	2	hardy	hardy	ADJ
asir-4475	34	3	’s	’s	PART
asir-4475	34	4	theorem	theorem	VERB
asir-4475	34	5	,	,	PUNCT
asir-4475	34	6	1914	1914	NUM
asir-4475	34	7	]	]	PUNCT
asir-4475	34	8	that	that	PRON
asir-4475	34	9	proved	prove	VERB
asir-4475	34	10	that	that	SCONJ
asir-4475	34	11	“	"	PUNCT
asir-4475	34	12	there	there	PRON
asir-4475	34	13	are	be	VERB
asir-4475	34	14	infinitely	infinitely	ADV
asir-4475	34	15	many	many	ADJ
asir-4475	34	16	zeros	zero	NOUN
asir-4475	34	17	of	of	ADP
asir-4475	34	18	(s	(s	NOUN
asir-4475	34	19	)	)	PUNCT
asir-4475	34	20	on	on	ADP
asir-4475	34	21	the	the	DET
asir-4475	34	22	critical	critical	ADJ
asir-4475	34	23	line	line	NOUN
asir-4475	34	24	”	"	PUNCT
asir-4475	34	25	is	be	AUX
asir-4475	34	26	not	not	PART
asir-4475	34	27	enough	enough	ADJ
asir-4475	34	28	.	.	PUNCT
asir-4475	35	1	all	all	DET
asir-4475	35	2	the	the	DET
asir-4475	35	3	nontrivial	nontrivial	ADJ
asir-4475	35	4	zeros	zero	NOUN
asir-4475	35	5	must	must	AUX
asir-4475	35	6	be	be	AUX
asir-4475	35	7	on	on	ADP
asir-4475	35	8	the	the	DET
asir-4475	35	9	critical	critical	ADJ
asir-4475	35	10	line	line	NOUN
asir-4475	35	11	,	,	PUNCT
asir-4475	35	12	if	if	SCONJ
asir-4475	35	13	one	one	PRON
asir-4475	35	14	wants	want	VERB
asir-4475	35	15	to	to	PART
asir-4475	35	16	prove	prove	VERB
asir-4475	35	17	rc	rc	PROPN
asir-4475	35	18	.	.	PUNCT
asir-4475	36	1	the	the	DET
asir-4475	36	2	author	author	NOUN
asir-4475	36	3	,	,	PUNCT
asir-4475	36	4	fausto	fausto	PROPN
asir-4475	36	5	galetto	galetto	NOUN
asir-4475	36	6	,	,	PUNCT
asir-4475	36	7	is	be	AUX
asir-4475	36	8	aware	aware	ADJ
asir-4475	36	9	that	that	SCONJ
asir-4475	36	10	(	(	PUNCT
asir-4475	36	11	in	in	ADP
asir-4475	36	12	these	these	DET
asir-4475	36	13	weeks	week	NOUN
asir-4475	36	14	)	)	PUNCT
asir-4475	36	15	he	he	PRON
asir-4475	36	16	has	have	AUX
asir-4475	36	17	been	be	AUX
asir-4475	36	18	affording	afford	VERB
asir-4475	36	19	a	a	DET
asir-4475	36	20	very	very	ADV
asir-4475	36	21	important	important	ADJ
asir-4475	36	22	problem	problem	NOUN
asir-4475	36	23	that	that	SCONJ
asir-4475	36	24	great	great	ADJ
asir-4475	36	25	mathematicians	mathematician	NOUN
asir-4475	36	26	have	have	AUX
asir-4475	36	27	failed	fail	VERB
asir-4475	36	28	to	to	PART
asir-4475	36	29	prove	prove	VERB
asir-4475	36	30	.	.	PUNCT
asir-4475	37	1	2	2	X
asir-4475	37	2	.	.	X
asir-4475	37	3	the	the	DET
asir-4475	37	4	“	"	PUNCT
asir-4475	37	5	transfer	transfer	NOUN
asir-4475	37	6	function	function	NOUN
asir-4475	37	7	”	"	PUNCT
asir-4475	37	8	and	and	CCONJ
asir-4475	37	9	the	the	DET
asir-4475	37	10	spira	spira	PROPN
asir-4475	37	11	criterion	criterion	NOUN
asir-4475	37	12	from	from	ADP
asir-4475	37	13	(	(	PUNCT
asir-4475	37	14	1	1	X
asir-4475	37	15	)	)	PUNCT
asir-4475	37	16	we	we	PRON
asir-4475	37	17	derive	derive	VERB
asir-4475	37	18	(1	(1	VERB
asir-4475	37	19	−	−	NOUN
asir-4475	37	20	s	s	PART
asir-4475	37	21	)	)	PUNCT
asir-4475	37	22	=	=	PUNCT
asir-4475	38	1	π0.5−𝑠	π0.5−𝑠	PROPN
asir-4475	38	2	γ(𝑠/2	γ(𝑠/2	NOUN
asir-4475	38	3	)	)	PUNCT
asir-4475	38	4	γ[(1−𝑠)/2	γ[(1−𝑠)/2	NOUN
asir-4475	38	5	]	]	PUNCT
asir-4475	38	6	)	)	PUNCT
asir-4475	38	7	(s	(s	NOUN
asir-4475	38	8	)	)	PUNCT
asir-4475	38	9	(	(	PUNCT
asir-4475	38	10	2a	2a	NUM
asir-4475	38	11	)	)	PUNCT
asir-4475	38	12	which	which	PRON
asir-4475	38	13	can	can	AUX
asir-4475	38	14	be	be	AUX
asir-4475	38	15	written	write	VERB
asir-4475	38	16	,	,	PUNCT
asir-4475	38	17	in	in	ADP
asir-4475	38	18	the	the	DET
asir-4475	38	19	form	form	NOUN
asir-4475	38	20	used	use	VERB
asir-4475	38	21	in	in	ADP
asir-4475	38	22	electronics	electronic	NOUN
asir-4475	38	23	,	,	PUNCT
asir-4475	38	24	communication	communication	NOUN
asir-4475	38	25	theory	theory	NOUN
asir-4475	38	26	and	and	CCONJ
asir-4475	38	27	control	control	NOUN
asir-4475	38	28	systems	system	NOUN
asir-4475	38	29	,	,	PUNCT
asir-4475	38	30	(1	(1	VERB
asir-4475	38	31	−	−	NOUN
asir-4475	38	32	s	s	PART
asir-4475	38	33	)	)	PUNCT
asir-4475	38	34	=	=	SYM
asir-4475	38	35	h(s)(s	h(s)(s	NOUN
asir-4475	38	36	)	)	PUNCT
asir-4475	38	37	(	(	PUNCT
asir-4475	38	38	2b	2b	NOUN
asir-4475	38	39	)	)	PUNCT
asir-4475	38	40	where	where	SCONJ
asir-4475	38	41	h(s	h(	NOUN
asir-4475	38	42	)	)	PUNCT
asir-4475	38	43	is	be	AUX
asir-4475	38	44	the	the	DET
asir-4475	38	45	“	"	PUNCT
asir-4475	38	46	transfer	transfer	NOUN
asir-4475	38	47	function	function	NOUN
asir-4475	38	48	”	"	PUNCT
asir-4475	38	49	that	that	PRON
asir-4475	38	50	links	link	VERB
asir-4475	38	51	the	the	DET
asir-4475	38	52	input	input	NOUN
asir-4475	38	53	(s	(s	NOUN
asir-4475	38	54	)	)	PUNCT
asir-4475	38	55	to	to	ADP
asir-4475	38	56	the	the	DET
asir-4475	38	57	output	output	NOUN
asir-4475	38	58	(1	(1	VERB
asir-4475	38	59	−	−	NOUN
asir-4475	38	60	s	s	NOUN
asir-4475	38	61	)	)	PUNCT
asir-4475	38	62	.	.	PUNCT
asir-4475	39	1	the	the	DET
asir-4475	39	2	“	"	PUNCT
asir-4475	39	3	transfer	transfer	NOUN
asir-4475	39	4	function	function	NOUN
asir-4475	39	5	”	"	PUNCT
asir-4475	39	6	is	be	AUX
asir-4475	39	7	the	the	DET
asir-4475	39	8	ratio	ratio	NOUN
asir-4475	39	9	output	output	NOUN
asir-4475	39	10	/	/	SYM
asir-4475	39	11	input	input	NOUN
asir-4475	39	12	.	.	PUNCT
asir-4475	40	1	when	when	SCONJ
asir-4475	40	2	=0	=0	ADP
asir-4475	40	3	the	the	DET
asir-4475	40	4	“	"	PUNCT
asir-4475	40	5	transfer	transfer	NOUN
asir-4475	40	6	function	function	NOUN
asir-4475	40	7	”	"	PUNCT
asir-4475	40	8	depends	depend	VERB
asir-4475	40	9	only	only	ADV
asir-4475	40	10	on	on	ADP
asir-4475	40	11	the	the	DET
asir-4475	40	12	“	"	PUNCT
asir-4475	40	13	frequency	frequency	NOUN
asir-4475	40	14	”	"	PUNCT
asir-4475	40	15			PROPN
asir-4475	40	16	;	;	PUNCT
asir-4475	40	17	from	from	ADP
asir-4475	40	18	now	now	ADV
asir-4475	40	19	on	on	ADV
asir-4475	40	20	we	we	PRON
asir-4475	40	21	use	use	VERB
asir-4475	40	22	the	the	DET
asir-4475	40	23	symbol	symbol	NOUN
asir-4475	40	24	s=+j	s=+j	ADV
asir-4475	40	25	,	,	PUNCT
asir-4475	40	26	as	as	SCONJ
asir-4475	40	27	it	it	PRON
asir-4475	40	28	is	be	AUX
asir-4475	40	29	customary	customary	ADJ
asir-4475	40	30	in	in	ADP
asir-4475	40	31	electronics	electronic	NOUN
asir-4475	40	32	,	,	PUNCT
asir-4475	40	33	communication	communication	NOUN
asir-4475	40	34	theory	theory	NOUN
asir-4475	40	35	and	and	CCONJ
asir-4475	40	36	control	control	NOUN
asir-4475	40	37	systems	system	NOUN
asir-4475	40	38	:	:	PUNCT
asir-4475	41	1	(1	(1	VERB
asir-4475	41	2	−	−	PROPN
asir-4475	42	1	σ	σ	NOUN
asir-4475	42	2	−	−	PROPN
asir-4475	42	3	jω	jω	NOUN
asir-4475	42	4	)	)	PUNCT
asir-4475	42	5	=	=	SYM
asir-4475	42	6	h(σ	h(σ	PROPN
asir-4475	42	7	+	+	CCONJ
asir-4475	42	8	jω)(σ	jω)(σ	NOUN
asir-4475	43	1	+	+	CCONJ
asir-4475	43	2	jω	jω	X
asir-4475	43	3	)	)	PUNCT
asir-4475	43	4	(	(	PUNCT
asir-4475	43	5	2c	2c	NOUN
asir-4475	43	6	)	)	PUNCT
asir-4475	43	7	when	when	SCONJ
asir-4475	43	8	=0	=0	PRON
asir-4475	43	9	we	we	PRON
asir-4475	43	10	have	have	AUX
asir-4475	43	11	(1	(1	VERB
asir-4475	43	12	−	−	NOUN
asir-4475	43	13	jω	jω	NOUN
asir-4475	43	14	)	)	PUNCT
asir-4475	43	15	=	=	SYM
asir-4475	43	16	h(jω)(jω	h(jω)(jω	PROPN
asir-4475	43	17	)	)	PUNCT
asir-4475	43	18	(	(	PUNCT
asir-4475	43	19	2d	2d	NOUN
asir-4475	43	20	)	)	PUNCT
asir-4475	43	21	squaring	square	VERB
asir-4475	43	22	both	both	DET
asir-4475	43	23	side	side	NOUN
asir-4475	43	24	and	and	CCONJ
asir-4475	43	25	taking	take	VERB
asir-4475	43	26	the	the	DET
asir-4475	43	27	absolute	absolute	ADJ
asir-4475	43	28	value	value	NOUN
asir-4475	43	29	,	,	PUNCT
asir-4475	43	30	we	we	PRON
asir-4475	43	31	get	get	VERB
asir-4475	43	32	|(1	|(1	NOUN
asir-4475	43	33	−	−	NOUN
asir-4475	43	34	jω)|2	jω)|2	PROPN
asir-4475	43	35	=	=	PUNCT
asir-4475	43	36	|h(jω)|2|(jω)|2	|h(jω)|2|(jω)|2	PROPN
asir-4475	43	37	(	(	PUNCT
asir-4475	43	38	3	3	X
asir-4475	43	39	)	)	PUNCT
asir-4475	43	40	this	this	DET
asir-4475	43	41	formula	formula	NOUN
asir-4475	43	42	is	be	AUX
asir-4475	43	43	the	the	DET
asir-4475	43	44	same	same	ADJ
asir-4475	43	45	as	as	ADP
asir-4475	43	46	the	the	DET
asir-4475	43	47	following	follow	VERB
asir-4475	43	48	s𝑂𝑢𝑡𝑝𝑢𝑡(jω	s𝑂𝑢𝑡𝑝𝑢𝑡(jω	NOUN
asir-4475	43	49	)	)	PUNCT
asir-4475	44	1	=	=	SYM
asir-4475	44	2	|h(jω)|2s𝐼𝑛𝑝𝑢𝑡(jω	|h(jω)|2s𝐼𝑛𝑝𝑢𝑡(jω	NUM
asir-4475	44	3	)	)	PUNCT
asir-4475	44	4	(	(	PUNCT
asir-4475	44	5	3b	3b	NUM
asir-4475	44	6	)	)	PUNCT
asir-4475	44	7	used	use	VERB
asir-4475	44	8	in	in	ADP
asir-4475	44	9	electronics	electronic	NOUN
asir-4475	44	10	,	,	PUNCT
asir-4475	44	11	communication	communication	NOUN
asir-4475	44	12	theory	theory	NOUN
asir-4475	44	13	and	and	CCONJ
asir-4475	44	14	control	control	NOUN
asir-4475	44	15	systems	system	NOUN
asir-4475	44	16	,	,	PUNCT
asir-4475	44	17	where	where	SCONJ
asir-4475	44	18	the	the	DET
asir-4475	44	19	function	function	NOUN
asir-4475	44	20	s(j	s(j	PROPN
asir-4475	44	21	)	)	PUNCT
asir-4475	44	22	is	be	AUX
asir-4475	44	23	the	the	DET
asir-4475	44	24	“	"	PUNCT
asir-4475	44	25	power	power	NOUN
asir-4475	44	26	spectral	spectral	ADJ
asir-4475	44	27	density	density	NOUN
asir-4475	44	28	”	"	PUNCT
asir-4475	44	29	.	.	PUNCT
asir-4475	45	1	we	we	PRON
asir-4475	45	2	name	name	VERB
asir-4475	45	3	here	here	ADV
asir-4475	45	4	|h(jω)|2	|h(jω)|2	VERB
asir-4475	45	5	the	the	DET
asir-4475	45	6	“	"	PUNCT
asir-4475	45	7	power	power	NOUN
asir-4475	45	8	transfer	transfer	NOUN
asir-4475	45	9	function	function	NOUN
asir-4475	45	10	”	"	PUNCT
asir-4475	45	11	.	.	PUNCT
asir-4475	46	1	in	in	ADP
asir-4475	46	2	analogy	analogy	NOUN
asir-4475	46	3	to	to	ADP
asir-4475	46	4	(	(	PUNCT
asir-4475	46	5	3b	3b	NUM
asir-4475	46	6	)	)	PUNCT
asir-4475	46	7	we	we	PRON
asir-4475	46	8	define	define	VERB
asir-4475	46	9	|(1	|(1	VERB
asir-4475	46	10	−	−	NOUN
asir-4475	46	11	s)|2	s)|2	NOUN
asir-4475	46	12	=	=	SYM
asir-4475	46	13	|h(s)|2|(s)|2	|h(s)|2|(s)|2	NOUN
asir-4475	46	14	(	(	PUNCT
asir-4475	46	15	3c	3c	NUM
asir-4475	46	16	)	)	PUNCT
asir-4475	46	17	and	and	CCONJ
asir-4475	46	18	name	name	NOUN
asir-4475	46	19	here	here	ADV
asir-4475	46	20	|h(s)|2	|h(s)|2	PUNCT
asir-4475	46	21	the	the	DET
asir-4475	46	22	“	"	PUNCT
asir-4475	46	23	generalised	generalised	ADJ
asir-4475	46	24	power	power	NOUN
asir-4475	46	25	transfer	transfer	NOUN
asir-4475	46	26	function	function	NOUN
asir-4475	46	27	”	"	PUNCT
asir-4475	46	28	.	.	PUNCT
asir-4475	47	1	the	the	DET
asir-4475	47	2	spira	spira	PROPN
asir-4475	47	3	criterion	criterion	NOUN
asir-4475	47	4	(	(	PUNCT
asir-4475	47	5	broughan	broughan	ADP
asir-4475	47	6	,	,	PUNCT
asir-4475	47	7	2017	2017	NUM
asir-4475	47	8	)	)	PUNCT
asir-4475	47	9	for	for	ADP
asir-4475	47	10	rc	rc	PROPN
asir-4475	47	11	is	be	AUX
asir-4475	47	12	the	the	DET
asir-4475	47	13	riemann	riemann	PROPN
asir-4475	47	14	hypothesis	hypothesis	NOUN
asir-4475	47	15	is	be	AUX
asir-4475	47	16	equivalent	equivalent	ADJ
asir-4475	47	17	to	to	ADP
asir-4475	47	18	the	the	DET
asir-4475	47	19	statement	statement	NOUN
asir-4475	47	20	that	that	PRON
asir-4475	47	21	|(1	|(1	VERB
asir-4475	47	22	−	−	PROPN
asir-4475	47	23	𝑠)|	𝑠)|	NOUN
asir-4475	47	24	>	>	X
asir-4475	47	25	|(𝑠)|	|(𝑠)|	NOUN
asir-4475	47	26	for	for	ADP
asir-4475	47	27	any	any	DET
asir-4475	47	28	0.5<<1	0.5<<1	NOUN
asir-4475	47	29	and	and	CCONJ
asir-4475	47	30	any	any	DET
asir-4475	47	31	𝜔	𝜔	PRON
asir-4475	47	32	≥	≥	NOUN
asir-4475	47	33	2𝜋	2𝜋	NOUN
asir-4475	47	34	+	+	CCONJ
asir-4475	47	35	0.1	0.1	NUM
asir-4475	47	36	.	.	PUNCT
asir-4475	48	1	notice	notice	VERB
asir-4475	48	2	that	that	SCONJ
asir-4475	48	3	|h(s)|	|h(s)|	PROPN
asir-4475	48	4	>	>	X
asir-4475	48	5	0	0	PUNCT
asir-4475	49	1	for	for	ADP
asir-4475	49	2	any	any	DET
asir-4475	49	3	s	s	NOUN
asir-4475	49	4	and	and	CCONJ
asir-4475	49	5	|h(0.5	|h(0.5	NOUN
asir-4475	49	6	+	+	CCONJ
asir-4475	49	7	jω)|	jω)|	PROPN
asir-4475	49	8	=	=	SYM
asir-4475	49	9	1	1	NUM
asir-4475	49	10	because	because	SCONJ
asir-4475	49	11	|h(1	|h(1	VERB
asir-4475	50	1	−	−	PROPN
asir-4475	50	2	s)h(s)|	s)h(s)|	NOUN
asir-4475	50	3	=	=	NOUN
asir-4475	50	4	1	1	NUM
asir-4475	50	5	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-4475	50	6	applied	apply	VERB
asir-4475	50	7	science	science	NOUN
asir-4475	50	8	and	and	CCONJ
asir-4475	50	9	innovative	innovative	ADJ
asir-4475	50	10	research	research	NOUN
asir-4475	50	11	vol	vol	NOUN
asir-4475	50	12	.	.	PROPN
asir-4475	51	1	6	6	NUM
asir-4475	51	2	,	,	PUNCT
asir-4475	51	3	no	no	INTJ
asir-4475	51	4	.	.	NOUN
asir-4475	51	5	1	1	NUM
asir-4475	51	6	,	,	PUNCT
asir-4475	51	7	2022	2022	NUM
asir-4475	51	8	17	17	NUM
asir-4475	51	9	published	publish	VERB
asir-4475	51	10	by	by	ADP
asir-4475	51	11	scholink	scholink	PROPN
asir-4475	51	12	inc	inc	PROPN
asir-4475	51	13	.	.	PROPN
asir-4475	52	1	since	since	SCONJ
asir-4475	52	2	|(1	|(1	VERB
asir-4475	52	3	−	−	PROPN
asir-4475	52	4	𝑠)/(𝑠)|	𝑠)/(𝑠)|	NOUN
asir-4475	52	5	=	=	SYM
asir-4475	52	6	|h(s)|	|h(s)|	PROPN
asir-4475	52	7	,	,	PUNCT
asir-4475	52	8	the	the	DET
asir-4475	52	9	spira	spira	PROPN
asir-4475	52	10	criterion	criterion	NOUN
asir-4475	52	11	(	(	PUNCT
asir-4475	52	12	broughan	broughan	ADP
asir-4475	52	13	,	,	PUNCT
asir-4475	52	14	2017	2017	NUM
asir-4475	52	15	)	)	PUNCT
asir-4475	52	16	for	for	ADP
asir-4475	52	17	rc	rc	PROPN
asir-4475	52	18	states	state	NOUN
asir-4475	52	19	that	that	SCONJ
asir-4475	52	20	|h(σ	|h(σ	PROPN
asir-4475	52	21	+	+	CCONJ
asir-4475	52	22	jω)|	jω)|	PROPN
asir-4475	52	23	>	>	X
asir-4475	52	24	1	1	NUM
asir-4475	52	25	for	for	ADP
asir-4475	52	26	any	any	DET
asir-4475	52	27	0.5<<1	0.5<<1	NOUN
asir-4475	52	28	and	and	CCONJ
asir-4475	52	29	any	any	DET
asir-4475	52	30	𝜔	𝜔	PRON
asir-4475	52	31	≥	≥	NOUN
asir-4475	52	32	2𝜋	2𝜋	NOUN
asir-4475	52	33	+	+	CCONJ
asir-4475	52	34	0.1	0.1	NUM
asir-4475	52	35	.	.	PUNCT
asir-4475	53	1	we	we	PRON
asir-4475	53	2	can	can	AUX
asir-4475	53	3	use	use	VERB
asir-4475	53	4	the	the	DET
asir-4475	53	5	infinite	infinite	ADJ
asir-4475	53	6	product	product	NOUN
asir-4475	53	7	expansion	expansion	NOUN
asir-4475	53	8	for	for	ADP
asir-4475	53	9	the	the	DET
asir-4475	53	10	gamma	gamma	NOUN
asir-4475	53	11	function	function	NOUN
asir-4475	53	12	and	and	CCONJ
asir-4475	53	13	we	we	PRON
asir-4475	53	14	get	get	VERB
asir-4475	53	15	|h(s)|	|h(s)|	PROPN
asir-4475	53	16	=	=	PROPN
asir-4475	53	17	|𝜋0.5−𝑠|	|𝜋0.5−𝑠|	PROPN
asir-4475	53	18	[	[	PUNCT
asir-4475	53	19	|γ	|γ	PROPN
asir-4475	53	20	(	(	PUNCT
asir-4475	53	21	𝑠	𝑠	PROPN
asir-4475	53	22	2	2	NUM
asir-4475	53	23	)	)	PUNCT
asir-4475	53	24	|	|	ADV
asir-4475	53	25	|γ	|γ	ADV
asir-4475	53	26	(	(	PUNCT
asir-4475	53	27	1−𝑠	1−𝑠	NUM
asir-4475	53	28	2	2	NUM
asir-4475	53	29	)	)	PUNCT
asir-4475	53	30	|	|	ADV
asir-4475	53	31	]	]	PUNCT
asir-4475	53	32	=	=	PUNCT
asir-4475	53	33	|𝜋0.5−𝑠||𝑒𝛾(0.5−𝑠)|	|𝜋0.5−𝑠||𝑒𝛾(0.5−𝑠)|	X
asir-4475	54	1	|	|	NOUN
asir-4475	54	2	1−𝑠	1−𝑠	NUM
asir-4475	54	3	𝑠	𝑠	PROPN
asir-4475	54	4	|∏	|∏	PROPN
asir-4475	54	5	{	{	PUNCT
asir-4475	54	6	|(1	|(1	PROPN
asir-4475	54	7	+	+	PROPN
asir-4475	54	8	1−𝑠	1−𝑠	PROPN
asir-4475	54	9	2𝑛	2𝑛	NOUN
asir-4475	54	10	)	)	PUNCT
asir-4475	55	1	|	|	ADV
asir-4475	55	2	|(1	|(1	PROPN
asir-4475	55	3	+	+	PROPN
asir-4475	55	4	𝑠	𝑠	PROPN
asir-4475	55	5	2𝑛	2𝑛	PROPN
asir-4475	55	6	)	)	PUNCT
asir-4475	55	7	|	|	ADV
asir-4475	55	8	|𝑒	|𝑒	VERB
asir-4475	55	9	𝑠−0.5	𝑠−0.5	NUM
asir-4475	55	10	𝑛	𝑛	PRON
asir-4475	55	11	|}∞	|}∞	NOUN
asir-4475	55	12	1	1	NUM
asir-4475	55	13	=	=	SYM
asir-4475	55	14	|𝜋0.5−𝑠||𝑒𝛾(0.5−𝑠)|	|𝜋0.5−𝑠||𝑒𝛾(0.5−𝑠)|	X
asir-4475	55	15	|	|	NOUN
asir-4475	55	16	1−𝑠	1−𝑠	NUM
asir-4475	55	17	𝑠	𝑠	PROPN
asir-4475	55	18	|∏	|∏	NOUN
asir-4475	55	19	{	{	PUNCT
asir-4475	55	20	|(2𝑛+1−𝑠)|	|(2𝑛+1−𝑠)|	NOUN
asir-4475	55	21	|(2𝑛+𝑠)|	|(2𝑛+𝑠)|	ADJ
asir-4475	55	22	|𝑒	|𝑒	PUNCT
asir-4475	55	23	𝑠−0.5	𝑠−0.5	NUM
asir-4475	55	24	𝑛	𝑛	PRON
asir-4475	55	25	|}∞	|}∞	NOUN
asir-4475	55	26	1	1	NUM
asir-4475	55	27	(	(	PUNCT
asir-4475	55	28	4	4	NUM
asir-4475	55	29	)	)	PUNCT
asir-4475	55	30	figure	figure	NOUN
asir-4475	55	31	2	2	NUM
asir-4475	55	32	depicts	depict	VERB
asir-4475	55	33	one	one	NUM
asir-4475	55	34	of	of	ADP
asir-4475	55	35	the	the	DET
asir-4475	55	36	“	"	PUNCT
asir-4475	55	37	products	product	NOUN
asir-4475	55	38	”	"	PUNCT
asir-4475	55	39	within	within	ADP
asir-4475	55	40	the	the	DET
asir-4475	55	41	braces	brace	NOUN
asir-4475	55	42	figure	figure	NOUN
asir-4475	55	43	2	2	NUM
asir-4475	55	44	.	.	PUNCT
asir-4475	55	45	picture	picture	NOUN
asir-4475	55	46	of	of	ADP
asir-4475	55	47	one	one	NUM
asir-4475	55	48	of	of	ADP
asir-4475	55	49	the	the	DET
asir-4475	55	50	“	"	PUNCT
asir-4475	55	51	products	product	NOUN
asir-4475	55	52	”	"	PUNCT
asir-4475	55	53	within	within	ADP
asir-4475	55	54	the	the	DET
asir-4475	55	55	braces	brace	NOUN
asir-4475	55	56	we	we	PRON
asir-4475	55	57	can	can	AUX
asir-4475	55	58	also	also	ADV
asir-4475	55	59	find	find	VERB
asir-4475	55	60	the	the	DET
asir-4475	55	61	“	"	PUNCT
asir-4475	55	62	generalised	generalised	ADJ
asir-4475	55	63	power	power	NOUN
asir-4475	55	64	transfer	transfer	NOUN
asir-4475	55	65	function	function	NOUN
asir-4475	55	66	”	"	PUNCT
asir-4475	55	67	|h(s)|2	|h(s)|2	PUNCT
asir-4475	55	68	=	=	SYM
asir-4475	55	69	|𝜋0.5−𝑠|2|[𝑒−𝛾(𝑠−0.5)]|	|𝜋0.5−𝑠|2|[𝑒−𝛾(𝑠−0.5)]|	NUM
asir-4475	55	70	2	2	NUM
asir-4475	55	71	|	|	NOUN
asir-4475	55	72	1−𝑠	1−𝑠	NUM
asir-4475	56	1	𝑠	𝑠	NOUN
asir-4475	56	2	|	|	ADV
asir-4475	56	3	2	2	NUM
asir-4475	56	4	∏	∏	PROPN
asir-4475	56	5	{	{	PUNCT
asir-4475	56	6	|(2𝑛+1−𝑠)|2	|(2𝑛+1−𝑠)|2	NOUN
asir-4475	56	7	|(2𝑛+𝑠)|2	|(2𝑛+𝑠)|2	VERB
asir-4475	56	8	|𝑒	|𝑒	PRON
asir-4475	56	9	𝑠−0.5	𝑠−0.5	NUM
asir-4475	56	10	𝑛	𝑛	PRON
asir-4475	56	11	|	|	ADJ
asir-4475	56	12	2	2	NUM
asir-4475	56	13	}	}	PUNCT
asir-4475	56	14	∞	∞	NUM
asir-4475	56	15	1	1	NUM
asir-4475	56	16	(	(	PUNCT
asir-4475	56	17	5	5	NUM
asir-4475	56	18	)	)	PUNCT
asir-4475	56	19	this	this	PRON
asir-4475	56	20	can	can	AUX
asir-4475	56	21	be	be	AUX
asir-4475	56	22	written	write	VERB
asir-4475	56	23	as	as	ADP
asir-4475	56	24	a	a	DET
asir-4475	56	25	real	real	ADJ
asir-4475	56	26	function	function	NOUN
asir-4475	56	27	a(,	a(,	VERB
asir-4475	56	28	)	)	PUNCT
asir-4475	57	1	a(σ	a(σ	PROPN
asir-4475	57	2	,	,	PUNCT
asir-4475	57	3	ω	ω	NOUN
asir-4475	57	4	)	)	PUNCT
asir-4475	57	5	=	=	SYM
asir-4475	57	6	𝜋1−2𝜎𝑒−𝛾(2𝜎−1	𝜋1−2𝜎𝑒−𝛾(2𝜎−1	PROPN
asir-4475	57	7	)	)	PUNCT
asir-4475	57	8	(	(	PUNCT
asir-4475	57	9	1−𝜎)2+𝜔2	1−𝜎)2+𝜔2	PROPN
asir-4475	57	10	𝜎2+𝜔2	𝜎2+𝜔2	PROPN
asir-4475	57	11	∏	∏	PROPN
asir-4475	57	12	{	{	PUNCT
asir-4475	57	13	(	(	PUNCT
asir-4475	57	14	2𝑛+1−𝜎)2+𝜔2	2𝑛+1−𝜎)2+𝜔2	NUM
asir-4475	57	15	(	(	PUNCT
asir-4475	57	16	2𝑛+𝜎)2+𝜔2	2𝑛+𝜎)2+𝜔2	NUM
asir-4475	57	17	𝑒	𝑒	X
asir-4475	57	18	(	(	PUNCT
asir-4475	57	19	2𝜎−1	2𝜎−1	NUM
asir-4475	57	20	)	)	PUNCT
asir-4475	57	21	𝑛	𝑛	PRON
asir-4475	57	22	}	}	PUNCT
asir-4475	57	23	∞	∞	NUM
asir-4475	57	24	1	1	NUM
asir-4475	57	25	(	(	PUNCT
asir-4475	57	26	5b	5b	NUM
asir-4475	57	27	)	)	PUNCT
asir-4475	57	28	3	3	NUM
asir-4475	57	29	.	.	PUNCT
asir-4475	58	1	the	the	DET
asir-4475	58	2	proof	proof	NOUN
asir-4475	58	3	of	of	ADP
asir-4475	58	4	rc	rc	PROPN
asir-4475	58	5	to	to	PART
asir-4475	58	6	prove	prove	VERB
asir-4475	58	7	rc	rc	PROPN
asir-4475	58	8	,	,	PUNCT
asir-4475	58	9	we	we	PRON
asir-4475	58	10	use	use	VERB
asir-4475	58	11	the	the	DET
asir-4475	58	12	spira	spira	PROPN
asir-4475	58	13	criterion	criterion	NOUN
asir-4475	58	14	(	(	PUNCT
asir-4475	58	15	broughan	broughan	ADP
asir-4475	58	16	,	,	PUNCT
asir-4475	58	17	2017	2017	NUM
asir-4475	58	18	)	)	PUNCT
asir-4475	58	19	for	for	ADP
asir-4475	58	20	rc	rc	PROPN
asir-4475	58	21	:	:	PUNCT
asir-4475	58	22	we	we	PRON
asir-4475	58	23	show	show	VERB
asir-4475	58	24	that	that	SCONJ
asir-4475	58	25	|h(s)|	|h(s)|	PROPN
asir-4475	58	26	=	=	PROPN
asir-4475	58	27	|h(σ	|h(σ	PROPN
asir-4475	58	28	+	+	X
asir-4475	58	29	jω)|	jω)|	PROPN
asir-4475	58	30	>	>	X
asir-4475	58	31	1	1	NUM
asir-4475	58	32	for	for	ADP
asir-4475	58	33	any	any	DET
asir-4475	58	34	0.5<<1	0.5<<1	NOUN
asir-4475	58	35	and	and	CCONJ
asir-4475	58	36	any	any	DET
asir-4475	58	37	ω	ω	NUM
asir-4475	58	38	≥	≥	NOUN
asir-4475	58	39	2π	2π	NOUN
asir-4475	58	40	+	+	X
asir-4475	58	41	0.1	0.1	NUM
asir-4475	58	42	.	.	PUNCT
asir-4475	59	1	for	for	ADP
asir-4475	59	2	any	any	DET
asir-4475	59	3	chosen	choose	VERB
asir-4475	59	4	𝜔	𝜔	ADP
asir-4475	59	5	≥	≥	NOUN
asir-4475	59	6	2𝜋	2𝜋	NOUN
asir-4475	59	7	+	+	CCONJ
asir-4475	59	8	0.1	0.1	NUM
asir-4475	59	9	,	,	PUNCT
asir-4475	59	10	we	we	PRON
asir-4475	59	11	can	can	AUX
asir-4475	59	12	find	find	VERB
asir-4475	59	13	a	a	DET
asir-4475	59	14	value	value	NOUN
asir-4475	59	15	n0	n0	NOUN
asir-4475	59	16	such	such	ADJ
asir-4475	59	17	that	that	SCONJ
asir-4475	59	18	,	,	PUNCT
asir-4475	59	19	for	for	ADP
asir-4475	59	20	any	any	DET
asir-4475	59	21	n	n	CCONJ
asir-4475	59	22	>	>	X
asir-4475	59	23	n0	n0	PROPN
asir-4475	59	24	,	,	PUNCT
asir-4475	59	25	the	the	DET
asir-4475	59	26	logarithm	logarithm	NOUN
asir-4475	59	27	is	be	AUX
asir-4475	59	28	expended	expend	VERB
asir-4475	59	29	as	as	ADP
asir-4475	59	30	ln	ln	ADJ
asir-4475	59	31	[	[	X
asir-4475	59	32	1	1	NUM
asir-4475	59	33	+	+	NUM
asir-4475	59	34	𝑧/(2𝑛	𝑧/(2𝑛	NOUN
asir-4475	59	35	)	)	PUNCT
asir-4475	59	36	]	]	PUNCT
asir-4475	60	1	≈	≈	PROPN
asir-4475	60	2	{	{	PUNCT
asir-4475	60	3	𝑧/(2𝑛	𝑧/(2𝑛	NOUN
asir-4475	60	4	)	)	PUNCT
asir-4475	60	5	−	−	PROPN
asir-4475	60	6	𝑧2/[2(2𝑛)2	𝑧2/[2(2𝑛)2	PROPN
asir-4475	60	7	]	]	PUNCT
asir-4475	60	8	}	}	PUNCT
asir-4475	60	9	,	,	PUNCT
asir-4475	60	10	with	with	ADP
asir-4475	60	11	𝑧	𝑧	PRON
asir-4475	60	12	=	=	SYM
asir-4475	60	13	σ	σ	PROPN
asir-4475	60	14	+	+	PROPN
asir-4475	60	15	jω	jω	PROPN
asir-4475	60	16	;	;	PUNCT
asir-4475	60	17	then	then	ADV
asir-4475	60	18	𝑙𝑛	𝑙𝑛	X
asir-4475	60	19	(	(	PUNCT
asir-4475	60	20	1	1	NUM
asir-4475	60	21	+	+	SYM
asir-4475	60	22	1	1	NUM
asir-4475	60	23	−	−	NUM
asir-4475	60	24	𝑠	𝑠	PROPN
asir-4475	60	25	2𝑛	2𝑛	PROPN
asir-4475	60	26	)	)	PUNCT
asir-4475	61	1	−	−	PROPN
asir-4475	61	2	𝑙𝑛	𝑙𝑛	NOUN
asir-4475	61	3	(	(	PUNCT
asir-4475	61	4	1	1	NUM
asir-4475	61	5	+	+	NUM
asir-4475	61	6	𝑠	𝑠	PROPN
asir-4475	61	7	2𝑛	2𝑛	PROPN
asir-4475	61	8	)	)	PUNCT
asir-4475	62	1	≈	≈	PROPN
asir-4475	62	2	(	(	PUNCT
asir-4475	62	3	1	1	NUM
asir-4475	62	4	−	−	PRON
asir-4475	62	5	2𝑠	2𝑠	NUM
asir-4475	62	6	)	)	PUNCT
asir-4475	62	7	2𝑛	2𝑛	PROPN
asir-4475	63	1	[	[	X
asir-4475	63	2	1	1	NUM
asir-4475	63	3	−	−	NUM
asir-4475	63	4	1	1	NUM
asir-4475	63	5	2(2𝑛	2(2𝑛	NUM
asir-4475	63	6	)	)	PUNCT
asir-4475	63	7	]	]	PUNCT
asir-4475	63	8	therefore	therefore	ADV
asir-4475	63	9	,	,	PUNCT
asir-4475	63	10	for	for	ADP
asir-4475	63	11	any	any	DET
asir-4475	63	12	n	n	CCONJ
asir-4475	63	13	>	>	X
asir-4475	63	14	n0	n0	X
asir-4475	63	15	,	,	PUNCT
asir-4475	63	16	each	each	DET
asir-4475	63	17	factor	factor	NOUN
asir-4475	63	18	in	in	ADP
asir-4475	63	19	the	the	DET
asir-4475	63	20	“	"	PUNCT
asir-4475	63	21	infinite	infinite	ADJ
asir-4475	63	22	product	product	NOUN
asir-4475	63	23	”	"	PUNCT
asir-4475	63	24	is	be	AUX
asir-4475	63	25	𝑎𝑛	𝑎𝑛	NOUN
asir-4475	63	26	=	=	PUNCT
asir-4475	63	27	{	{	PUNCT
asir-4475	63	28	|1	|1	PUNCT
asir-4475	64	1	+	+	CCONJ
asir-4475	64	2	(	(	PUNCT
asir-4475	64	3	1	1	NUM
asir-4475	64	4	−	−	NOUN
asir-4475	64	5	𝑠)/(2𝑛)|	𝑠)/(2𝑛)|	NOUN
asir-4475	64	6	|1	|1	X
asir-4475	65	1	+	+	NUM
asir-4475	65	2	𝑠/(2𝑛)|	𝑠/(2𝑛)|	NOUN
asir-4475	65	3	|𝑒	|𝑒	VERB
asir-4475	65	4	𝑠−0.5	𝑠−0.5	NUM
asir-4475	65	5	𝑛	𝑛	PRON
asir-4475	65	6	|	|	NOUN
asir-4475	65	7	}	}	PUNCT
asir-4475	65	8	≈	≈	PROPN
asir-4475	65	9	|𝑒	|𝑒	X
asir-4475	65	10	(	(	PUNCT
asir-4475	65	11	𝑠−0.5	𝑠−0.5	NUM
asir-4475	65	12	)	)	PUNCT
asir-4475	65	13	[	[	PUNCT
asir-4475	65	14	1	1	NUM
asir-4475	65	15	(	(	PUNCT
asir-4475	65	16	2𝑛)2	2𝑛)2	NUM
asir-4475	65	17	]	]	PUNCT
asir-4475	66	1	|	|	ADV
asir-4475	66	2	=	=	PUNCT
asir-4475	66	3	𝑒	𝑒	PROPN
asir-4475	66	4	𝑅𝑒{(𝑠−0.5	𝑅𝑒{(𝑠−0.5	PROPN
asir-4475	66	5	)	)	PUNCT
asir-4475	66	6	[	[	PUNCT
asir-4475	66	7	1	1	NUM
asir-4475	66	8	(	(	PUNCT
asir-4475	66	9	2𝑛)2	2𝑛)2	NUM
asir-4475	66	10	]	]	PUNCT
asir-4475	66	11	}	}	PUNCT
asir-4475	66	12	=	=	SYM
asir-4475	66	13	𝑒	𝑒	PROPN
asir-4475	66	14	(	(	PUNCT
asir-4475	66	15	𝜎−0.5	𝜎−0.5	NUM
asir-4475	66	16	)	)	PUNCT
asir-4475	66	17	[	[	PUNCT
asir-4475	66	18	1	1	NUM
asir-4475	66	19	(	(	PUNCT
asir-4475	66	20	2𝑛)2	2𝑛)2	NUM
asir-4475	66	21	]	]	PUNCT
asir-4475	66	22	>	>	X
asir-4475	66	23	1	1	NUM
asir-4475	66	24	points	point	NOUN
asir-4475	66	25	of	of	ADP
asir-4475	66	26	the	the	DET
asir-4475	66	27	zeta	zeta	NOUN
asir-4475	66	28	function	function	NOUN
asir-4475	66	29	0	0	NUM
asir-4475	66	30	0.5	0.5	NUM
asir-4475	66	31	1	1	NUM
asir-4475	66	32	re(s	re(s	ADJ
asir-4475	66	33	)	)	PUNCT
asir-4475	66	34	im(s	im(	NOUN
asir-4475	66	35	)	)	PUNCT
asir-4475	66	36	1	1	NUM
asir-4475	66	37	-	-	PUNCT
asir-4475	66	38	s	s	X
asir-4475	66	39	-2n	-2n	PROPN
asir-4475	66	40	2n+s	2n+s	PROPN
asir-4475	66	41	2n+1	2n+1	PROPN
asir-4475	66	42	-	-	SYM
asir-4475	66	43	s	s	PART
asir-4475	66	44	s	s	NOUN
asir-4475	66	45	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-4475	66	46	applied	apply	VERB
asir-4475	66	47	science	science	NOUN
asir-4475	66	48	and	and	CCONJ
asir-4475	66	49	innovative	innovative	ADJ
asir-4475	66	50	research	research	NOUN
asir-4475	66	51	vol	vol	NOUN
asir-4475	66	52	.	.	PROPN
asir-4475	67	1	6	6	NUM
asir-4475	67	2	,	,	PUNCT
asir-4475	67	3	no	no	INTJ
asir-4475	67	4	.	.	NOUN
asir-4475	67	5	1	1	NUM
asir-4475	67	6	,	,	PUNCT
asir-4475	67	7	2022	2022	NUM
asir-4475	67	8	18	18	NUM
asir-4475	67	9	published	publish	VERB
asir-4475	67	10	by	by	ADP
asir-4475	67	11	scholink	scholink	PROPN
asir-4475	67	12	inc	inc	PROPN
asir-4475	67	13	.	.	PROPN
asir-4475	67	14	see	see	VERB
asir-4475	67	15	figure	figure	NOUN
asir-4475	67	16	3	3	NUM
asir-4475	67	17	figure	figure	NOUN
asir-4475	67	18	3	3	NUM
asir-4475	67	19	.	.	PUNCT
asir-4475	68	1	picture	picture	NOUN
asir-4475	68	2	of	of	ADP
asir-4475	68	3	the	the	DET
asir-4475	68	4	first	first	ADJ
asir-4475	68	5	factor	factor	NOUN
asir-4475	68	6	within	within	ADP
asir-4475	68	7	the	the	DET
asir-4475	68	8	braces	brace	NOUN
asir-4475	68	9	since	since	SCONJ
asir-4475	68	10	𝑎𝑛	𝑎𝑛	PROPN
asir-4475	68	11	>	>	SYM
asir-4475	68	12	𝑎𝑛+1	𝑎𝑛+1	PROPN
asir-4475	68	13	,	,	PUNCT
asir-4475	68	14	we	we	PRON
asir-4475	68	15	get	get	VERB
asir-4475	68	16	∏	∏	NUM
asir-4475	68	17	𝑎𝑛	𝑎𝑛	NOUN
asir-4475	68	18	∞	∞	NUM
asir-4475	68	19	1	1	NUM
asir-4475	68	20	>	>	SYM
asir-4475	68	21	1	1	NUM
asir-4475	68	22	and	and	CCONJ
asir-4475	68	23	hence	hence	ADV
asir-4475	68	24	|h(s)|	|h(s)|	PROPN
asir-4475	68	25	>	>	X
asir-4475	68	26	1	1	X
asir-4475	68	27	.	.	PUNCT
asir-4475	69	1	therefore	therefore	ADV
asir-4475	69	2	rc	rc	PROPN
asir-4475	69	3	is	be	AUX
asir-4475	69	4	proved	prove	VERB
asir-4475	69	5	by	by	ADP
asir-4475	69	6	the	the	DET
asir-4475	69	7	spira	spira	PROPN
asir-4475	69	8	criterion	criterion	NOUN
asir-4475	69	9	(	(	PUNCT
asir-4475	69	10	broughan	broughan	ADP
asir-4475	69	11	,	,	PUNCT
asir-4475	69	12	2017	2017	NUM
asir-4475	69	13	)	)	PUNCT
asir-4475	69	14	.	.	PUNCT
asir-4475	70	1	figure	figure	VERB
asir-4475	70	2	4	4	NUM
asir-4475	70	3	.	.	PUNCT
asir-4475	71	1	behaviour	behaviour	NOUN
asir-4475	71	2	of	of	ADP
asir-4475	71	3	|𝐇(𝛔	|𝐇(𝛔	PROPN
asir-4475	71	4	+	+	CCONJ
asir-4475	71	5	𝐣𝛚)|	𝐣𝛚)|	PROPN
asir-4475	71	6	,	,	PUNCT
asir-4475	71	7	abscissa	abscissa	PROPN
asir-4475	71	8	is	be	AUX
asir-4475	71	9	𝛔	𝛔	ADP
asir-4475	71	10	,	,	PUNCT
asir-4475	71	11	curves	curve	NOUN
asir-4475	71	12	are	be	AUX
asir-4475	71	13	indexed	index	VERB
asir-4475	71	14	by	by	ADP
asir-4475	71	15	𝛚	𝛚	PROPN
asir-4475	71	16	0	0	NUM
asir-4475	71	17	0.5	0.5	NUM
asir-4475	71	18	1	1	NUM
asir-4475	71	19	re(s	re(s	ADJ
asir-4475	71	20	)	)	PUNCT
asir-4475	71	21	im(s	im(	NOUN
asir-4475	71	22	)	)	PUNCT
asir-4475	71	23	1	1	NUM
asir-4475	71	24	-	-	SYM
asir-4475	71	25	s	s	X
asir-4475	71	26	s	s	NOUN
asir-4475	71	27	1+s/2s/2	1+s/2s/2	NUM
asir-4475	71	28	(	(	PUNCT
asir-4475	71	29	1	1	NUM
asir-4475	71	30	-	-	PUNCT
asir-4475	71	31	s)/2	s)/2	ADJ
asir-4475	71	32	1.5	1.5	NUM
asir-4475	71	33	s-0.5	s-0.5	NOUN
asir-4475	71	34	+	+	CCONJ
asir-4475	71	35	−	−	PROPN
asir-4475	71	36	+	+	CCONJ
asir-4475	71	37	−	−	PROPN
asir-4475	71	38	.	.	PUNCT
asir-4475	72	1	for	for	ADP
asir-4475	72	2	n=1	n=1	ADP
asir-4475	72	3	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	NOUN
asir-4475	72	4	applied	apply	VERB
asir-4475	72	5	science	science	NOUN
asir-4475	72	6	and	and	CCONJ
asir-4475	72	7	innovative	innovative	ADJ
asir-4475	72	8	research	research	NOUN
asir-4475	72	9	vol	vol	NOUN
asir-4475	72	10	.	.	PROPN
asir-4475	73	1	6	6	NUM
asir-4475	73	2	,	,	PUNCT
asir-4475	73	3	no	no	INTJ
asir-4475	73	4	.	.	NOUN
asir-4475	73	5	1	1	NUM
asir-4475	73	6	,	,	PUNCT
asir-4475	73	7	2022	2022	NUM
asir-4475	73	8	19	19	NUM
asir-4475	73	9	published	publish	VERB
asir-4475	73	10	by	by	ADP
asir-4475	73	11	scholink	scholink	PROPN
asir-4475	73	12	inc	inc	PROPN
asir-4475	73	13	.	.	PROPN
asir-4475	73	14	figure	figure	NOUN
asir-4475	73	15	4	4	NUM
asir-4475	73	16	show	show	VERB
asir-4475	73	17	the	the	DET
asir-4475	73	18	behaviour	behaviour	NOUN
asir-4475	73	19	of	of	ADP
asir-4475	73	20	|h(σ	|h(σ	PROPN
asir-4475	73	21	+	+	CCONJ
asir-4475	74	1	jω)|	jω)|	PROPN
asir-4475	74	2	for	for	ADP
asir-4475	74	3	various	various	ADJ
asir-4475	74	4	values	value	NOUN
asir-4475	74	5	of	of	ADP
asir-4475	74	6	(	(	PUNCT
asir-4475	74	7	abscissa	abscissa	ADJ
asir-4475	74	8	)	)	PUNCT
asir-4475	74	9	σ	σ	NOUN
asir-4475	74	10	and	and	CCONJ
asir-4475	74	11	various	various	ADJ
asir-4475	74	12	ω	ω	PROPN
asir-4475	74	13	(	(	PUNCT
asir-4475	74	14	curves	curve	NOUN
asir-4475	74	15	)	)	PUNCT
asir-4475	74	16	;	;	PUNCT
asir-4475	74	17	all	all	DET
asir-4475	74	18	curves	curve	NOUN
asir-4475	74	19	intersects	intersect	NOUN
asir-4475	74	20	at	at	ADP
asir-4475	74	21	σ=0.5	σ=0.5	NOUN
asir-4475	74	22	and	and	CCONJ
asir-4475	74	23	have	have	AUX
asir-4475	74	24	ordinate	ordinate	VERB
asir-4475	74	25	|h(σ	|h(σ	PROPN
asir-4475	74	26	+	+	CCONJ
asir-4475	74	27	jω)|	jω)|	PROPN
asir-4475	74	28	=	=	SYM
asir-4475	74	29	1	1	NUM
asir-4475	74	30	we	we	PRON
asir-4475	74	31	can	can	AUX
asir-4475	74	32	confirm	confirm	VERB
asir-4475	74	33	the	the	DET
asir-4475	74	34	previous	previous	ADJ
asir-4475	74	35	result	result	NOUN
asir-4475	74	36	by	by	ADP
asir-4475	74	37	writing	write	VERB
asir-4475	74	38	it	it	PRON
asir-4475	74	39	in	in	ADP
asir-4475	74	40	a	a	DET
asir-4475	74	41	different	different	ADJ
asir-4475	74	42	way	way	NOUN
asir-4475	74	43	(	(	PUNCT
asir-4475	74	44	abramowitz	abramowitz	PROPN
asir-4475	74	45	et	et	PROPN
asir-4475	74	46	al	al	PROPN
asir-4475	74	47	.	.	PROPN
asir-4475	74	48	,	,	PUNCT
asir-4475	74	49	1965	1965	NUM
asir-4475	74	50	)	)	PUNCT
asir-4475	74	51	the	the	DET
asir-4475	74	52	gamma	gamma	PROPN
asir-4475	74	53	function	function	NOUN
asir-4475	74	54	;	;	PUNCT
asir-4475	74	55	so	so	CCONJ
asir-4475	74	56	the	the	DET
asir-4475	74	57	absolute	absolute	ADJ
asir-4475	74	58	“	"	PUNCT
asir-4475	74	59	transfer	transfer	NOUN
asir-4475	74	60	function	function	NOUN
asir-4475	74	61	”	"	PUNCT
asir-4475	74	62	becomes	become	VERB
asir-4475	74	63	|h(s)|	|h(s)|	PROPN
asir-4475	74	64	=	=	PROPN
asir-4475	74	65	|𝜋0.5−𝑠|	|𝜋0.5−𝑠|	ADP
asir-4475	74	66	|γ	|γ	ADV
asir-4475	74	67	(	(	PUNCT
asir-4475	74	68	𝜎	𝜎	PROPN
asir-4475	74	69	2	2	NUM
asir-4475	74	70	)	)	PUNCT
asir-4475	74	71	|	|	ADV
asir-4475	74	72	|γ	|γ	ADV
asir-4475	74	73	(	(	PUNCT
asir-4475	74	74	1−𝜎	1−𝜎	NUM
asir-4475	74	75	2	2	NUM
asir-4475	74	76	)	)	PUNCT
asir-4475	74	77	|	|	ADV
asir-4475	74	78	√∏	√∏	X
asir-4475	75	1	[	[	X
asir-4475	75	2	1	1	NUM
asir-4475	75	3	+	+	CCONJ
asir-4475	75	4	(	(	PUNCT
asir-4475	75	5	𝜔/2	𝜔/2	PUNCT
asir-4475	75	6	𝑛+(1−𝜎)/2	𝑛+(1−𝜎)/2	PROPN
asir-4475	75	7	)	)	PUNCT
asir-4475	75	8	2	2	NUM
asir-4475	75	9	]	]	PUNCT
asir-4475	75	10	/	/	PUNCT
asir-4475	76	1	[	[	X
asir-4475	76	2	1	1	NUM
asir-4475	76	3	+	+	CCONJ
asir-4475	76	4	(	(	PUNCT
asir-4475	76	5	𝜔/2	𝜔/2	X
asir-4475	76	6	𝑛+𝜎/2	𝑛+𝜎/2	ADV
asir-4475	76	7	)	)	PUNCT
asir-4475	76	8	2	2	NUM
asir-4475	76	9	]	]	SYM
asir-4475	76	10	∞	∞	NUM
asir-4475	76	11	0	0	NUM
asir-4475	76	12	(	(	PUNCT
asir-4475	76	13	4b	4b	X
asir-4475	76	14	)	)	PUNCT
asir-4475	76	15	let	let	VERB
asir-4475	76	16	’s	’s	NOUN
asir-4475	76	17	now	now	ADV
asir-4475	76	18	consider	consider	VERB
asir-4475	76	19	the	the	DET
asir-4475	76	20	“	"	PUNCT
asir-4475	76	21	generalised	generalised	ADJ
asir-4475	76	22	power	power	NOUN
asir-4475	76	23	transfer	transfer	NOUN
asir-4475	76	24	function	function	NOUN
asir-4475	76	25	”	"	PUNCT
asir-4475	76	26	|h(s)|2	|h(s)|2	ADP
asir-4475	76	27	;	;	PUNCT
asir-4475	76	28	from	from	ADP
asir-4475	76	29	above	above	ADV
asir-4475	76	30	,	,	PUNCT
asir-4475	76	31	we	we	PRON
asir-4475	76	32	have	have	VERB
asir-4475	76	33	the	the	DET
asir-4475	76	34	power	power	NOUN
asir-4475	76	35	difference	difference	NOUN
asir-4475	76	36	|(1	|(1	VERB
asir-4475	76	37	−	−	PROPN
asir-4475	76	38	s)|2	s)|2	NOUN
asir-4475	76	39	−	−	NOUN
asir-4475	76	40	|(s)|2	|(s)|2	NOUN
asir-4475	76	41	=	=	SYM
asir-4475	76	42	|(s)|2{|h(s)|2	|(s)|2{|h(s)|2	NOUN
asir-4475	76	43	−	−	NOUN
asir-4475	76	44	1	1	NUM
asir-4475	76	45	}	}	PUNCT
asir-4475	76	46	>	>	X
asir-4475	76	47	0	0	X
asir-4475	76	48	.	.	PUNCT
asir-4475	77	1	let	let	VERB
asir-4475	77	2	’s	’s	NOUN
asir-4475	77	3	consider	consider	VERB
asir-4475	77	4	the	the	DET
asir-4475	77	5	equation	equation	NOUN
asir-4475	77	6	|(s)|2{|h(s)|2	|(s)|2{|h(s)|2	NOUN
asir-4475	77	7	−	−	NOUN
asir-4475	77	8	1	1	X
asir-4475	77	9	}	}	PUNCT
asir-4475	77	10	=	=	SYM
asir-4475	77	11	0	0	NUM
asir-4475	77	12	(	(	PUNCT
asir-4475	77	13	6	6	NUM
asir-4475	77	14	)	)	PUNCT
asir-4475	77	15	form	form	NOUN
asir-4475	77	16	(	(	PUNCT
asir-4475	77	17	6	6	NUM
asir-4475	77	18	)	)	PUNCT
asir-4475	77	19	we	we	PRON
asir-4475	77	20	find	find	VERB
asir-4475	77	21	two	two	NUM
asir-4475	77	22	solutions	solution	NOUN
asir-4475	77	23	|(s)|2	|(s)|2	PUNCT
asir-4475	77	24	=	=	SYM
asir-4475	77	25	0	0	NUM
asir-4475	77	26	(	(	PUNCT
asir-4475	77	27	7	7	NUM
asir-4475	77	28	)	)	PUNCT
asir-4475	77	29	{	{	PUNCT
asir-4475	77	30	|h(s)|2	|h(s)|2	X
asir-4475	77	31	−	−	PROPN
asir-4475	77	32	1	1	NUM
asir-4475	77	33	}	}	PUNCT
asir-4475	77	34	=	=	SYM
asir-4475	77	35	0	0	PUNCT
asir-4475	78	1	(	(	PUNCT
asir-4475	78	2	8)	8)	NUM
asir-4475	78	3	(	(	PUNCT
asir-4475	78	4	7	7	NUM
asir-4475	78	5	)	)	PUNCT
asir-4475	78	6	entail	entail	NOUN
asir-4475	78	7	that	that	PRON
asir-4475	78	8	s	s	VERB
asir-4475	78	9	=	=	SYM
asir-4475	78	10	σ	σ	PROPN
asir-4475	79	1	+	+	CCONJ
asir-4475	79	2	jω	jω	X
asir-4475	79	3	is	be	AUX
asir-4475	79	4	a	a	DET
asir-4475	79	5	zero	zero	NUM
asir-4475	79	6	of	of	ADP
asir-4475	79	7	the	the	DET
asir-4475	79	8	zeta	zeta	NOUN
asir-4475	79	9	function	function	NOUN
asir-4475	79	10	(σ	(σ	ADV
asir-4475	79	11	+	+	X
asir-4475	79	12	jω	jω	NOUN
asir-4475	79	13	)	)	PUNCT
asir-4475	79	14	,	,	PUNCT
asir-4475	79	15	while	while	SCONJ
asir-4475	79	16	(	(	PUNCT
asir-4475	79	17	8)	8)	NUM
asir-4475	79	18	entails	entail	VERB
asir-4475	79	19	that	that	SCONJ
asir-4475	79	20	the	the	DET
asir-4475	79	21	real	real	ADJ
asir-4475	79	22	function	function	NOUN
asir-4475	79	23	a(,	a(,	VERB
asir-4475	79	24	)	)	PUNCT
asir-4475	79	25	equals	equal	VERB
asir-4475	79	26	1	1	NUM
asir-4475	79	27	for	for	ADP
asir-4475	79	28	any	any	DET
asir-4475	79	29			PROPN
asir-4475	79	30	𝜋1−2𝜎𝑒−𝛾(2𝜎−1	𝜋1−2𝜎𝑒−𝛾(2𝜎−1	PROPN
asir-4475	79	31	)	)	PUNCT
asir-4475	79	32	(	(	PUNCT
asir-4475	79	33	1−𝜎)2+𝜔2	1−𝜎)2+𝜔2	PROPN
asir-4475	79	34	𝜎2+𝜔2	𝜎2+𝜔2	PROPN
asir-4475	79	35	∏	∏	PROPN
asir-4475	79	36	{	{	PUNCT
asir-4475	79	37	(	(	PUNCT
asir-4475	79	38	2𝑛+1−𝜎)2+𝜔2	2𝑛+1−𝜎)2+𝜔2	NUM
asir-4475	79	39	(	(	PUNCT
asir-4475	79	40	2𝑛+𝜎)2+𝜔2	2𝑛+𝜎)2+𝜔2	NUM
asir-4475	79	41	𝑒	𝑒	X
asir-4475	79	42	(	(	PUNCT
asir-4475	79	43	2𝜎−1	2𝜎−1	NUM
asir-4475	79	44	)	)	PUNCT
asir-4475	79	45	𝑛	𝑛	PRON
asir-4475	79	46	}	}	PUNCT
asir-4475	79	47	∞	∞	NOUN
asir-4475	79	48	1	1	NUM
asir-4475	79	49	=	=	SYM
asir-4475	79	50	1	1	NUM
asir-4475	79	51	(	(	PUNCT
asir-4475	79	52	9	9	NUM
asir-4475	79	53	)	)	PUNCT
asir-4475	79	54	it	it	PRON
asir-4475	79	55	is	be	AUX
asir-4475	79	56	easily	easily	ADV
asir-4475	79	57	seen	see	VERB
asir-4475	79	58	that	that	SCONJ
asir-4475	79	59	(	(	PUNCT
asir-4475	79	60	9	9	X
asir-4475	79	61	)	)	PUNCT
asir-4475	79	62	has	have	VERB
asir-4475	79	63	a	a	DET
asir-4475	79	64	unique	unique	ADJ
asir-4475	79	65	solution	solution	NOUN
asir-4475	79	66	𝜎	𝜎	NOUN
asir-4475	80	1	=	=	SYM
asir-4475	80	2	0.5	0.5	NUM
asir-4475	80	3	;	;	PUNCT
asir-4475	80	4	therefore	therefore	ADV
asir-4475	80	5	the	the	DET
asir-4475	80	6	zeros	zero	NOUN
asir-4475	80	7	of	of	ADP
asir-4475	80	8	the	the	DET
asir-4475	80	9	zeta	zeta	NOUN
asir-4475	80	10	function	function	NOUN
asir-4475	80	11	(σ	(σ	ADV
asir-4475	80	12	+	+	CCONJ
asir-4475	80	13	jω	jω	X
asir-4475	80	14	)	)	PUNCT
asir-4475	80	15	are	be	AUX
asir-4475	80	16	all	all	ADV
asir-4475	80	17	on	on	ADP
asir-4475	80	18	the	the	DET
asir-4475	80	19	critical	critical	ADJ
asir-4475	80	20	line	line	NOUN
asir-4475	80	21	𝑠	𝑠	NOUN
asir-4475	80	22	=	=	SYM
asir-4475	80	23	0.5	0.5	NUM
asir-4475	80	24	+	+	CCONJ
asir-4475	80	25	𝑗𝜔	𝑗𝜔	ADJ
asir-4475	80	26	;	;	PUNCT
asir-4475	80	27	this	this	PRON
asir-4475	80	28	is	be	AUX
asir-4475	80	29	the	the	DET
asir-4475	80	30	same	same	ADJ
asir-4475	80	31	result	result	NOUN
asir-4475	80	32	found	find	VERB
asir-4475	80	33	before	before	ADP
asir-4475	80	34	by	by	ADP
asir-4475	80	35	the	the	DET
asir-4475	80	36	spira	spira	PROPN
asir-4475	80	37	criterion	criterion	NOUN
asir-4475	80	38	.	.	PUNCT
asir-4475	81	1	we	we	PRON
asir-4475	81	2	can	can	AUX
asir-4475	81	3	confirm	confirm	VERB
asir-4475	81	4	the	the	DET
asir-4475	81	5	previous	previous	ADJ
asir-4475	81	6	result	result	NOUN
asir-4475	81	7	by	by	ADP
asir-4475	81	8	writing	write	VERB
asir-4475	81	9	in	in	ADP
asir-4475	81	10	a	a	DET
asir-4475	81	11	different	different	ADJ
asir-4475	81	12	way	way	NOUN
asir-4475	81	13	(	(	PUNCT
asir-4475	81	14	abramowitz	abramowitz	PROPN
asir-4475	81	15	et	et	PROPN
asir-4475	81	16	al	al	PROPN
asir-4475	81	17	.	.	PROPN
asir-4475	81	18	,	,	PUNCT
asir-4475	81	19	1965	1965	NUM
asir-4475	81	20	)	)	PUNCT
asir-4475	81	21	the	the	DET
asir-4475	81	22	gamma	gamma	PROPN
asir-4475	81	23	function	function	NOUN
asir-4475	81	24	;	;	PUNCT
asir-4475	81	25	so	so	CCONJ
asir-4475	81	26	the	the	DET
asir-4475	81	27	“	"	PUNCT
asir-4475	81	28	generalised	generalised	ADJ
asir-4475	81	29	power	power	NOUN
asir-4475	81	30	transfer	transfer	NOUN
asir-4475	81	31	function	function	NOUN
asir-4475	81	32	”	"	PUNCT
asir-4475	81	33	|h(s)|2	|h(s)|2	X
asir-4475	81	34	becomes	become	VERB
asir-4475	81	35	|h(s)|2	|h(s)|2	ADP
asir-4475	81	36	=	=	SYM
asir-4475	81	37	𝜋1−2𝜎	𝜋1−2𝜎	PROPN
asir-4475	81	38	[	[	PUNCT
asir-4475	81	39	|γ	|γ	PROPN
asir-4475	81	40	(	(	PUNCT
asir-4475	81	41	𝜎	𝜎	PROPN
asir-4475	81	42	2	2	NUM
asir-4475	81	43	)	)	PUNCT
asir-4475	81	44	|	|	ADV
asir-4475	81	45	|γ	|γ	ADV
asir-4475	81	46	(	(	PUNCT
asir-4475	81	47	1−𝜎	1−𝜎	NUM
asir-4475	81	48	2	2	NUM
asir-4475	81	49	)	)	PUNCT
asir-4475	81	50	|	|	CCONJ
asir-4475	81	51	]	]	PUNCT
asir-4475	81	52	2	2	NUM
asir-4475	81	53	∏	∏	X
asir-4475	82	1	[	[	X
asir-4475	82	2	1	1	NUM
asir-4475	82	3	+	+	CCONJ
asir-4475	82	4	(	(	PUNCT
asir-4475	82	5	𝜔/2	𝜔/2	PUNCT
asir-4475	82	6	𝑛+(1−𝜎)/2	𝑛+(1−𝜎)/2	PROPN
asir-4475	82	7	)	)	PUNCT
asir-4475	82	8	2	2	NUM
asir-4475	82	9	]	]	PUNCT
asir-4475	82	10	/	/	PUNCT
asir-4475	83	1	[	[	X
asir-4475	83	2	1	1	NUM
asir-4475	83	3	+	+	CCONJ
asir-4475	83	4	(	(	PUNCT
asir-4475	83	5	𝜔/2	𝜔/2	X
asir-4475	83	6	𝑛+𝜎/2	𝑛+𝜎/2	ADV
asir-4475	83	7	)	)	PUNCT
asir-4475	83	8	2	2	NUM
asir-4475	83	9	]	]	SYM
asir-4475	83	10	∞	∞	NUM
asir-4475	83	11	0	0	PUNCT
asir-4475	84	1	=	=	SYM
asir-4475	84	2	1	1	NUM
asir-4475	84	3	(	(	PUNCT
asir-4475	84	4	9b	9b	NUM
asir-4475	84	5	)	)	PUNCT
asir-4475	84	6	it	it	PRON
asir-4475	84	7	is	be	AUX
asir-4475	84	8	easily	easily	ADV
asir-4475	84	9	seen	see	VERB
asir-4475	84	10	that	that	SCONJ
asir-4475	84	11	(	(	PUNCT
asir-4475	84	12	9b	9b	NOUN
asir-4475	84	13	)	)	PUNCT
asir-4475	84	14	has	have	VERB
asir-4475	84	15	a	a	DET
asir-4475	84	16	unique	unique	ADJ
asir-4475	84	17	solution	solution	NOUN
asir-4475	84	18	𝜎	𝜎	NOUN
asir-4475	85	1	=	=	SYM
asir-4475	85	2	0.5	0.5	NUM
asir-4475	85	3	;	;	PUNCT
asir-4475	85	4	therefore	therefore	ADV
asir-4475	85	5	…	…	PUNCT
asir-4475	85	6	.	.	PUNCT
asir-4475	86	1	(	(	PUNCT
asir-4475	86	2	as	as	ADP
asir-4475	86	3	before	before	ADV
asir-4475	86	4	)	)	PUNCT
asir-4475	86	5	.	.	PUNCT
asir-4475	87	1	4	4	X
asir-4475	87	2	.	.	X
asir-4475	87	3	conclusion	conclusion	NOUN
asir-4475	87	4	titchmarsh	titchmarsh	NOUN
asir-4475	87	5	proved	prove	VERB
asir-4475	87	6	that	that	SCONJ
asir-4475	87	7	there	there	PRON
asir-4475	87	8	are	be	VERB
asir-4475	87	9	infinite	infinite	ADJ
asir-4475	87	10	zeros	zero	NOUN
asir-4475	87	11	of	of	ADP
asir-4475	87	12	the	the	DET
asir-4475	87	13	riemann	riemann	PROPN
asir-4475	87	14	zeta	zeta	PROPN
asir-4475	87	15	function	function	PROPN
asir-4475	87	16	(s	(s	NOUN
asir-4475	87	17	)	)	PUNCT
asir-4475	87	18	in	in	ADP
asir-4475	87	19	the	the	DET
asir-4475	87	20	critical	critical	ADJ
asir-4475	87	21	strip	strip	NOUN
asir-4475	87	22	,	,	PUNCT
asir-4475	87	23	that	that	ADV
asir-4475	87	24	is	be	AUX
asir-4475	87	25	there	there	PRON
asir-4475	87	26	are	be	VERB
asir-4475	87	27	infinite	infinite	ADJ
asir-4475	87	28	values	value	NOUN
asir-4475	87	29	sk=k+ik	sk=k+ik	ADJ
asir-4475	87	30	such	such	ADJ
asir-4475	87	31	that	that	SCONJ
asir-4475	87	32	(sk)=0	(sk)=0	NOUN
asir-4475	87	33	,	,	PUNCT
asir-4475	87	34	[	[	X
asir-4475	87	35	0	0	X
asir-4475	87	36	<	<	X
asir-4475	87	37	k	k	X
asir-4475	87	38	<	<	X
asir-4475	87	39	1	1	NUM
asir-4475	87	40	]	]	PUNCT
asir-4475	87	41	.	.	PUNCT
asir-4475	88	1	g.	g.	NOUN
asir-4475	88	2	hardy	hardy	ADJ
asir-4475	89	1	[	[	X
asir-4475	89	2	hardy	hardy	ADJ
asir-4475	89	3	’s	’s	PART
asir-4475	89	4	theorem	theorem	VERB
asir-4475	89	5	,	,	PUNCT
asir-4475	89	6	1914	1914	NUM
asir-4475	89	7	]	]	PUNCT
asir-4475	89	8	proved	prove	VERB
asir-4475	89	9	that	that	SCONJ
asir-4475	89	10	“	"	PUNCT
asir-4475	89	11	there	there	PRON
asir-4475	89	12	are	be	VERB
asir-4475	89	13	infinitely	infinitely	ADV
asir-4475	89	14	many	many	ADJ
asir-4475	89	15	zeros	zero	NOUN
asir-4475	89	16	of	of	ADP
asir-4475	89	17	(s	(s	NOUN
asir-4475	89	18	)	)	PUNCT
asir-4475	89	19	on	on	ADP
asir-4475	89	20	the	the	DET
asir-4475	89	21	critical	critical	ADJ
asir-4475	89	22	line	line	NOUN
asir-4475	89	23	”	"	PUNCT
asir-4475	89	24	:	:	PUNCT
asir-4475	89	25	that	that	DET
asir-4475	89	26	fact	fact	NOUN
asir-4475	89	27	was	be	AUX
asir-4475	89	28	not	not	PART
asir-4475	89	29	conclusive	conclusive	ADJ
asir-4475	89	30	,	,	PUNCT
asir-4475	89	31	because	because	SCONJ
asir-4475	89	32	all	all	DET
asir-4475	89	33	the	the	DET
asir-4475	89	34	zeros	zero	NOUN
asir-4475	89	35	must	must	AUX
asir-4475	89	36	be	be	AUX
asir-4475	89	37	on	on	ADP
asir-4475	89	38	the	the	DET
asir-4475	89	39	critical	critical	ADJ
asir-4475	89	40	line	line	NOUN
asir-4475	89	41	.	.	PUNCT
asir-4475	90	1	in	in	ADP
asir-4475	90	2	this	this	DET
asir-4475	90	3	paper	paper	NOUN
asir-4475	90	4	,	,	PUNCT
asir-4475	90	5	to	to	PART
asir-4475	90	6	prove	prove	VERB
asir-4475	90	7	rc	rc	PROPN
asir-4475	90	8	we	we	PRON
asir-4475	90	9	used	use	VERB
asir-4475	90	10	first	first	ADV
asir-4475	90	11	the	the	DET
asir-4475	90	12	“	"	PUNCT
asir-4475	90	13	spira	spira	PROPN
asir-4475	90	14	criterion	criterion	NOUN
asir-4475	90	15	”	"	PUNCT
asir-4475	91	1	[	[	X
asir-4475	91	2	with	with	ADP
asir-4475	91	3	the	the	DET
asir-4475	91	4	transfer	transfer	NOUN
asir-4475	91	5	function	function	NOUN
asir-4475	91	6	h(s	h(s	PROPN
asir-4475	91	7	)	)	PUNCT
asir-4475	91	8	]	]	PUNCT
asir-4475	91	9	and	and	CCONJ
asir-4475	91	10	second	second	ADJ
asir-4475	91	11	the	the	DET
asir-4475	91	12	“	"	PUNCT
asir-4475	91	13	generalised	generalised	ADJ
asir-4475	91	14	power	power	NOUN
asir-4475	91	15	transfer	transfer	NOUN
asir-4475	91	16	function	function	NOUN
asir-4475	91	17	”	"	PUNCT
asir-4475	91	18	|h(s)|2	|h(s)|2	NOUN
asir-4475	91	19	.	.	PUNCT
asir-4475	92	1	the	the	DET
asir-4475	92	2	result	result	NOUN
asir-4475	92	3	is	be	AUX
asir-4475	92	4	that	that	SCONJ
asir-4475	92	5	rh	rh	PROPN
asir-4475	92	6	(	(	PUNCT
asir-4475	92	7	rc	rc	PROPN
asir-4475	92	8	)	)	PUNCT
asir-4475	92	9	is	be	AUX
asir-4475	92	10	true	true	ADJ
asir-4475	92	11	.	.	PUNCT
asir-4475	93	1	(	(	PUNCT
asir-4475	93	2	as	as	SCONJ
asir-4475	93	3	done	do	VERB
asir-4475	93	4	in	in	ADP
asir-4475	93	5	previous	previous	ADJ
asir-4475	93	6	papers	paper	NOUN
asir-4475	93	7	,	,	PUNCT
asir-4475	93	8	fausto	fausto	PROPN
asir-4475	93	9	galetto	galetto	NOUN
asir-4475	93	10	,	,	PUNCT
asir-4475	93	11	2014	2014	NUM
asir-4475	93	12	,	,	PUNCT
asir-4475	93	13	2018	2018	NUM
asir-4475	93	14	,	,	PUNCT
asir-4475	93	15	2019	2019	NUM
asir-4475	93	16	)	)	PUNCT
asir-4475	93	17	.	.	PUNCT
asir-4475	94	1	www.scholink.org/ojs/index.php/asir	www.scholink.org/ojs/index.php/asir	PROPN
asir-4475	94	2	applied	apply	VERB
asir-4475	94	3	science	science	NOUN
asir-4475	94	4	and	and	CCONJ
asir-4475	94	5	innovative	innovative	ADJ
asir-4475	94	6	research	research	NOUN
asir-4475	94	7	vol	vol	NOUN
asir-4475	94	8	.	.	PROPN
asir-4475	95	1	6	6	NUM
asir-4475	95	2	,	,	PUNCT
asir-4475	95	3	no	no	INTJ
asir-4475	95	4	.	.	NOUN
asir-4475	95	5	1	1	NUM
asir-4475	95	6	,	,	PUNCT
asir-4475	95	7	2022	2022	NUM
asir-4475	95	8	20	20	NUM
asir-4475	95	9	published	publish	VERB
asir-4475	95	10	by	by	ADP
asir-4475	95	11	scholink	scholink	PROPN
asir-4475	95	12	inc	inc	PROPN
asir-4475	95	13	.	.	PROPN
asir-4475	95	14	references	reference	NOUN
asir-4475	95	15	abramowitz	abramowitz	PROPN
asir-4475	95	16	,	,	PUNCT
asir-4475	95	17	m.	m.	NOUN
asir-4475	95	18	,	,	PUNCT
asir-4475	95	19	&	&	CCONJ
asir-4475	95	20	stegun	stegun	PROPN
asir-4475	95	21	,	,	PUNCT
asir-4475	95	22	i.	i.	PROPN
asir-4475	95	23	a.	a.	PROPN
asir-4475	95	24	(	(	PUNCT
asir-4475	95	25	1965	1965	NUM
asir-4475	95	26	)	)	PUNCT
asir-4475	95	27	.	.	PUNCT
asir-4475	96	1	handbook	handbook	NOUN
asir-4475	96	2	of	of	ADP
asir-4475	96	3	mathematical	mathematical	ADJ
asir-4475	96	4	functions	function	NOUN
asir-4475	96	5	.	.	PUNCT
asir-4475	97	1	national	national	PROPN
asir-4475	97	2	bureau	bureau	PROPN
asir-4475	97	3	of	of	ADP
asir-4475	97	4	standards	standard	NOUN
asir-4475	97	5	,	,	PUNCT
asir-4475	97	6	applied	apply	VERB
asir-4475	97	7	math	math	NOUN
asir-4475	97	8	.	.	PUNCT
asir-4475	98	1	series	series	PROPN
asir-4475	98	2	#	#	SYM
asir-4475	98	3	55	55	NUM
asir-4475	98	4	,	,	PUNCT
asir-4475	98	5	dover	dover	PROPN
asir-4475	98	6	publications	publication	NOUN
asir-4475	98	7	,	,	PUNCT
asir-4475	98	8	sec	sec	PROPN
asir-4475	98	9	.	.	PROPN
asir-4475	98	10	6.5	6.5	NUM
asir-4475	98	11	.	.	PUNCT
asir-4475	99	1	bombieri	bombieri	PROPN
asir-4475	99	2	,	,	PUNCT
asir-4475	99	3	e.	e.	PROPN
asir-4475	99	4	(	(	PUNCT
asir-4475	99	5	2000	2000	NUM
asir-4475	99	6	)	)	PUNCT
asir-4475	99	7	.	.	PUNCT
asir-4475	100	1	problems	problem	NOUN
asir-4475	100	2	of	of	ADP
asir-4475	100	3	the	the	DET
asir-4475	100	4	millennium	millennium	NOUN
asir-4475	100	5	:	:	PUNCT
asir-4475	100	6	the	the	DET
asir-4475	100	7	riemann	riemann	PROPN
asir-4475	100	8	hypothesis	hypothesis	NOUN
asir-4475	100	9	.	.	PUNCT
asir-4475	101	1	retrieved	retrieve	VERB
asir-4475	101	2	from	from	ADP
asir-4475	101	3	http://claymath.org/prizeproblems/riemann.htm	http://claymath.org/prizeproblems/riemann.htm	PROPN
asir-4475	101	4	broughan	broughan	PROPN
asir-4475	101	5	,	,	PUNCT
asir-4475	101	6	k.	k.	PROPN
asir-4475	101	7	(	(	PUNCT
asir-4475	101	8	2017	2017	NUM
asir-4475	101	9	)	)	PUNCT
asir-4475	101	10	.	.	PUNCT
asir-4475	102	1	equivalents	equivalent	NOUN
asir-4475	102	2	of	of	ADP
asir-4475	102	3	the	the	DET
asir-4475	102	4	riemann	riemann	PROPN
asir-4475	102	5	hypothesis	hypothesis	NOUN
asir-4475	102	6	,	,	PUNCT
asir-4475	102	7	arithmetic	arithmetic	ADJ
asir-4475	102	8	equivalents	equivalent	NOUN
asir-4475	102	9	,	,	PUNCT
asir-4475	102	10	encyclopedia	encyclopedia	NOUN
asir-4475	102	11	of	of	ADP
asir-4475	102	12	mathematics	mathematic	NOUN
asir-4475	102	13	and	and	CCONJ
asir-4475	102	14	its	its	PRON
asir-4475	102	15	applications	application	NOUN
asir-4475	102	16	.	.	PUNCT
asir-4475	103	1	cambridge	cambridge	PROPN
asir-4475	103	2	university	university	PROPN
asir-4475	103	3	press	press	NOUN
asir-4475	103	4	.	.	PUNCT
asir-4475	104	1	fausto	fausto	PROPN
asir-4475	104	2	galetto	galetto	NOUN
asir-4475	104	3	.	.	PUNCT
asir-4475	105	1	(	(	PUNCT
asir-4475	105	2	2014	2014	NUM
asir-4475	105	3	)	)	PUNCT
asir-4475	105	4	.	.	PUNCT
asir-4475	106	1	riemann	riemann	PROPN
asir-4475	106	2	hypothesis	hypothesis	NOUN
asir-4475	106	3	proved	prove	VERB
asir-4475	106	4	.	.	PUNCT
asir-4475	107	1	academia	academia	PROPN
asir-4475	107	2	arena	arena	PROPN
asir-4475	107	3	,	,	PUNCT
asir-4475	107	4	6(12	6(12	NUM
asir-4475	107	5	)	)	PUNCT
asir-4475	107	6	,	,	PUNCT
asir-4475	107	7	19	19	NUM
asir-4475	107	8	-	-	SYM
asir-4475	107	9	22	22	NUM
asir-4475	107	10	.	.	PUNCT
asir-4475	108	1	fausto	fausto	NOUN
asir-4475	108	2	galetto	galetto	NOUN
asir-4475	108	3	.	.	PUNCT
asir-4475	109	1	(	(	PUNCT
asir-4475	109	2	2018	2018	NUM
asir-4475	109	3	)	)	PUNCT
asir-4475	109	4	.	.	PUNCT
asir-4475	110	1	a	a	DET
asir-4475	110	2	new	new	ADJ
asir-4475	110	3	proof	proof	NOUN
asir-4475	110	4	of	of	ADP
asir-4475	110	5	the	the	DET
asir-4475	110	6	riemann	riemann	PROPN
asir-4475	110	7	hypothesis	hypothesis	NOUN
asir-4475	110	8	.	.	PUNCT
asir-4475	111	1	research	research	NOUN
asir-4475	111	2	trends	trend	NOUN
asir-4475	111	3	on	on	ADP
asir-4475	111	4	mathematics	mathematic	NOUN
asir-4475	111	5	and	and	CCONJ
asir-4475	111	6	statistics	statistic	NOUN
asir-4475	111	7	,	,	PUNCT
asir-4475	111	8	3	3	NUM
asir-4475	111	9	,	,	PUNCT
asir-4475	111	10	23	23	NUM
asir-4475	111	11	-	-	SYM
asir-4475	111	12	35	35	NUM
asir-4475	111	13	,	,	PUNCT
asir-4475	111	14	2019	2019	NUM
asir-4475	111	15	and	and	CCONJ
asir-4475	111	16	hal	hal	PROPN
asir-4475	111	17	archive	archive	PROPN
asir-4475	111	18	,	,	PUNCT
asir-4475	111	19	2018	2018	NUM
asir-4475	111	20	.	.	PUNCT
asir-4475	112	1	fausto	fausto	NOUN
asir-4475	112	2	galetto	galetto	NOUN
asir-4475	112	3	.	.	PUNCT
asir-4475	113	1	(	(	PUNCT
asir-4475	113	2	2019	2019	NUM
asir-4475	113	3	)	)	PUNCT
asir-4475	113	4	.	.	PUNCT
asir-4475	114	1	riemann	riemann	PROPN
asir-4475	114	2	’s	’s	PART
asir-4475	114	3	conjecture	conjecture	NOUN
asir-4475	114	4	,	,	PUNCT
asir-4475	114	5	a	a	DET
asir-4475	114	6	“	"	PUNCT
asir-4475	114	7	one	one	NUM
asir-4475	114	8	page	page	NOUN
asir-4475	114	9	proof	proof	NOUN
asir-4475	114	10	(	(	PUNCT
asir-4475	114	11	new	new	ADJ
asir-4475	114	12	)	)	PUNCT
asir-4475	114	13	”	"	PUNCT
asir-4475	114	14	.	.	PUNCT
asir-4475	115	1	hal	hal	PROPN
asir-4475	115	2	archive	archive	PROPN
asir-4475	115	3	,	,	PUNCT
asir-4475	115	4	2019	2019	NUM
asir-4475	115	5	and	and	CCONJ
asir-4475	115	6	academia.edu	academia.edu	PROPN
asir-4475	115	7	2019	2019	NUM
asir-4475	115	8	.	.	PUNCT
asir-4475	116	1	luigi	luigi	PROPN
asir-4475	116	2	amerio	amerio	PROPN
asir-4475	116	3	.	.	PUNCT
asir-4475	117	1	(	(	PUNCT
asir-4475	117	2	1989	1989	NUM
asir-4475	117	3	)	)	PUNCT
asir-4475	117	4	.	.	PUNCT
asir-4475	118	1	analisi	analisi	PROPN
asir-4475	118	2	matematica	matematica	PROPN
asir-4475	118	3	,	,	PUNCT
asir-4475	118	4	iii	iii	PROPN
asir-4475	118	5	,	,	PUNCT
asir-4475	118	6	parte	parte	NOUN
asir-4475	118	7	1	1	NUM
asir-4475	118	8	°	°	PROPN
asir-4475	118	9	,	,	PUNCT
asir-4475	118	10	utet	utet	NOUN
asir-4475	118	11	.	.	PUNCT
asir-4475	118	12	titchmarsh	titchmarsh	PROPN
asir-4475	118	13	,	,	PUNCT
asir-4475	118	14	e.	e.	PROPN
asir-4475	118	15	c.	c.	PROPN
asir-4475	118	16	(	(	PUNCT
asir-4475	118	17	1986	1986	NUM
asir-4475	118	18	)	)	PUNCT
asir-4475	118	19	.	.	PUNCT
asir-4475	119	1	the	the	DET
asir-4475	119	2	theory	theory	NOUN
asir-4475	119	3	of	of	ADP
asir-4475	119	4	the	the	DET
asir-4475	119	5	riemann	riemann	PROPN
asir-4475	119	6	zeta	zeta	PROPN
asir-4475	119	7	-	-	PUNCT
asir-4475	119	8	function	function	NOUN
asir-4475	119	9	.	.	PUNCT
asir-4475	120	1	clarendon	clarendon	PROPN
asir-4475	120	2	press	press	PROPN
asir-4475	120	3	,	,	PUNCT
asir-4475	120	4	oxford	oxford	PROPN
asir-4475	120	5	.	.	PUNCT
asir-4475	121	1	wolfram	wolfram	PROPN
asir-4475	121	2	mathworld	mathworld	PROPN
asir-4475	121	3	,	,	PUNCT
asir-4475	121	4	xi	xi	ADP
asir-4475	121	5	function	function	NOUN
asir-4475	121	6	.	.	PUNCT
asir-4475	122	1	(	(	PUNCT
asir-4475	122	2	2019	2019	NUM
asir-4475	122	3	)	)	PUNCT
asir-4475	122	4	.	.	PUNCT