id	sid	eid	entity	type
ma-287	1	1	2025	CARDINAL
ma-287	2	1	j. math	PERSON
ma-287	4	1	5 (	CARDINAL
ma-287	4	2	12doi	CARDINAL
ma-287	4	3	10.28924	CARDINAL
ma-287	4	4	christophe chesneau department of mathematics	ORG
ma-287	4	5	lmno	GPE
ma-287	4	6	14032	DATE
ma-287	4	7	france	GPE
ma-287	7	1	srinivasa ramanujan	ORG
ma-287	12	1	1	CARDINAL
ma-287	13	1	13	CARDINAL
ma-287	14	1	taylor	PERSON
ma-287	15	1	∞∑ k=1	PERSON
ma-287	17	1	0	CARDINAL
ma-287	17	2	2	CARDINAL
ma-287	17	3	8	CARDINAL
ma-287	18	1	1− log(x	TIME
ma-287	19	1	1 k	QUANTITY
ma-287	19	2	1− x)k	TIME
ma-287	20	1	1	CARDINAL
ma-287	21	1	1	CARDINAL
ma-287	21	2	0	CARDINAL
ma-287	21	3	2	CARDINAL
ma-287	22	1	4	CARDINAL
ma-287	22	2	0	CARDINAL
ma-287	24	1	26	CARDINAL
ma-287	24	2	2024	DATE
ma-287	27	1	j. math	PERSON
ma-287	29	1	10.28924	CARDINAL
ma-287	29	2	2of	CARDINAL
ma-287	30	1	4	CARDINAL
ma-287	31	1	1− log(x	TIME
ma-287	32	1	1)2	MONEY
ma-287	32	2	0	CARDINAL
ma-287	33	1	log(x) ≤	QUANTITY
ma-287	34	1	0	CARDINAL
ma-287	34	2	0	CARDINAL
ma-287	34	3	2	CARDINAL
ma-287	37	1	srinivasa	ORG
ma-287	37	2	6= 1	DATE
ma-287	37	3	1 log(x)	TIME
ma-287	40	1	12	CARDINAL
ma-287	40	2	364	CARDINAL
ma-287	43	1	1/2)[1/(1 + √ x	DATE
ma-287	46	1	2	CARDINAL
ma-287	46	2	chapter 31	LAW
ma-287	46	3	29	CARDINAL
ma-287	46	4	5	CARDINAL
ma-287	47	1	1 x	QUANTITY
ma-287	47	2	1	CARDINAL
ma-287	50	1	1	CARDINAL
ma-287	51	1	φ(x −	GPE
ma-287	51	2	1)−	CARDINAL
ma-287	53	1	∞∑ k=1	PERSON
ma-287	53	2	2	CARDINAL
ma-287	53	3	2k−1(x2−k−1)2	CARDINAL
ma-287	57	1	4	CARDINAL
ma-287	61	1	1	CARDINAL
ma-287	61	2	1	CARDINAL
ma-287	62	1	3	CARDINAL
ma-287	62	2	6	DATE
ma-287	62	3	7	DATE
ma-287	62	4	9–11	CARDINAL
ma-287	62	5	16].some	ORG
ma-287	62	6	section 2	LAW
ma-287	67	1	section 3	LAW
ma-287	68	1	j. math	PERSON
ma-287	70	1	10.28924	CARDINAL
ma-287	71	1	4	CARDINAL
ma-287	71	2	2	CARDINAL
ma-287	72	1	2.1	CARDINAL
ma-287	74	1	1)−	CARDINAL
ma-287	74	2	φ	ORG
ma-287	74	3	2.1	CARDINAL
ma-287	76	1	φ	ORG
ma-287	77	1	φ(x − 1)−	GPE
ma-287	79	1	∞∑ k=1	PERSON
ma-287	80	1	2k−1(x2	CARDINAL
ma-287	82	1	− φ	PERSON
ma-287	82	2	2k(x2	CARDINAL
ma-287	84	1	1	CARDINAL
ma-287	84	2	φ(x −	GPE
ma-287	84	3	2n(x2	CARDINAL
ma-287	85	1	20(x2	CARDINAL
ma-287	85	2	1	CARDINAL
ma-287	86	1	2n(x2	CARDINAL
ma-287	87	1	n∑ k=1	PERSON
ma-287	87	2	2k−1(x2	CARDINAL
ma-287	89	1	− φ	PERSON
ma-287	89	2	2k(x2	CARDINAL
ma-287	89	3	n∑ k=1	PERSON
ma-287	90	1	2n(x2−n−1	CARDINAL
ma-287	90	2	2n(e2−n	CARDINAL
ma-287	90	3	log(x)−1	ORG
ma-287	91	1	2n	CARDINAL
ma-287	91	2	1 + 2−n log(x)]− 1	QUANTITY
ma-287	91	3	φ	ORG
ma-287	91	4	φ(x − 1)−	GPE
ma-287	92	1	1)−	ORDINAL
ma-287	92	2	2n(x2	CARDINAL
ma-287	94	1	2n(x2	CARDINAL
ma-287	95	1	n∑ k=1	PERSON
ma-287	97	1	∞∑ k=1	PERSON
ma-287	98	1	φ	ORG
ma-287	98	2	1)− +	DATE
ma-287	98	3	∞∑ k=1	PERSON
ma-287	101	1	j. math	PERSON
ma-287	103	1	10.28924	CARDINAL
ma-287	103	2	4	CARDINAL
ma-287	104	1	2.2	CARDINAL
ma-287	105	1	2.1	CARDINAL
ma-287	106	1	2.2	CARDINAL
ma-287	107	1	0	CARDINAL
ma-287	107	2	1− log(x	TIME
ma-287	109	1	4].(2	CARDINAL
ma-287	109	2	0	CARDINAL
ma-287	109	3	1 x	QUANTITY
ma-287	109	4	1	CARDINAL
ma-287	112	1	srinivasa ramanujan	ORG
ma-287	112	2	2	CARDINAL
ma-287	112	3	chapter 31	LAW
ma-287	112	4	399	CARDINAL
ma-287	112	5	5	CARDINAL
ma-287	113	1	eight	CARDINAL
ma-287	113	2	0	CARDINAL
ma-287	114	1	− 1)2	PERSON
ma-287	117	1	1)3(3 +	DATE
ma-287	118	1	4	CARDINAL
ma-287	118	2	1	CARDINAL
ma-287	119	1	1− √	TIME
ma-287	121	1	1)3/2√	CARDINAL
ma-287	123	1	5	CARDINAL
ma-287	123	2	1	CARDINAL
ma-287	124	1	1	CARDINAL
ma-287	128	1	6	CARDINAL
ma-287	128	2	1	CARDINAL
ma-287	131	1	∞∑ k=1	PERSON
ma-287	131	2	1 + x2	QUANTITY
ma-287	131	3	2	CARDINAL
ma-287	132	1	7	CARDINAL
ma-287	132	2	0	CARDINAL
ma-287	133	1	2 +∞∑ k=1 sin	QUANTITY
ma-287	133	2	2k−2(x2	CARDINAL
ma-287	134	1	2k−2(x2	CARDINAL
ma-287	134	2	1)(3 +	DATE
ma-287	135	1	0	CARDINAL
ma-287	135	2	cos(x	PERSON
ma-287	137	1	+∞∑ k=1 sin	TIME
ma-287	137	2	2k−2(x2	CARDINAL
ma-287	137	3	2k−2(x2	CARDINAL
ma-287	137	4	1)(3 +	DATE
ma-287	139	1	j. math	PERSON
ma-287	141	1	10.28924	CARDINAL
ma-287	141	2	5(9	CARDINAL
ma-287	141	3	0	CARDINAL
ma-287	141	4	1)−	ORDINAL
ma-287	142	1	2 +∞∑ k=1	QUANTITY
ma-287	142	2	2k−2(x2	CARDINAL
ma-287	142	3	2k−2(x2	CARDINAL
ma-287	142	4	1)(3 +	DATE
ma-287	143	1	10	CARDINAL
ma-287	143	2	0	CARDINAL
ma-287	143	3	1)−	ORDINAL
ma-287	144	1	2 +∞∑ k=1	QUANTITY
ma-287	144	2	2k−2(x2	CARDINAL
ma-287	144	3	2k−2(x2	CARDINAL
ma-287	144	4	1)(3 +	DATE
ma-287	146	1	0	CARDINAL
ma-287	146	2	2.1	CARDINAL
ma-287	147	1	1− log(x	TIME
ma-287	148	1	1)−	CARDINAL
ma-287	150	1	∞∑ k=1	PERSON
ma-287	151	1	2k−1(x2	CARDINAL
ma-287	153	1	− φ	PERSON
ma-287	153	2	2k(x2	CARDINAL
ma-287	154	1	2k−1(x2	CARDINAL
ma-287	156	1	1)−	ORDINAL
ma-287	156	2	2k(x2−k	CARDINAL
ma-287	156	3	2k−1(x2	CARDINAL
ma-287	158	1	2k(x2−k	CARDINAL
ma-287	158	2	2k−1(x2	CARDINAL
ma-287	158	3	1 + x2	QUANTITY
ma-287	159	1	2	CARDINAL
ma-287	159	2	2k−1(x2	CARDINAL
ma-287	161	1	4	CARDINAL
ma-287	162	1	1− log(x	TIME
ma-287	164	1	2	CARDINAL
ma-287	164	2	0	CARDINAL
ma-287	164	3	2.1	CARDINAL
ma-287	165	1	1 x	QUANTITY
ma-287	165	2	1	CARDINAL
ma-287	166	1	1)−	CARDINAL
ma-287	168	1	∞∑ k=1	PERSON
ma-287	169	1	2k−1(x2	CARDINAL
ma-287	171	1	− φ	PERSON
ma-287	171	2	2k(x2	CARDINAL
ma-287	172	1	1 2k−1(x2−(k−1	CARDINAL
ma-287	172	2	1	CARDINAL
ma-287	172	3	1 2k(x2	CARDINAL
ma-287	172	4	2k−1(x2	CARDINAL
ma-287	174	1	1)−	ORDINAL
ma-287	174	2	2k(x2−k	CARDINAL
ma-287	174	3	1	CARDINAL
ma-287	177	1	2k−1(x2	CARDINAL
ma-287	179	1	1)2	ORDINAL
ma-287	181	1	1)2	CARDINAL
ma-287	181	2	1	CARDINAL
ma-287	183	1	j. math	PERSON
ma-287	185	1	10.28924	CARDINAL
ma-287	186	1	6hence	CARDINAL
ma-287	186	2	1 x	QUANTITY
ma-287	186	3	1	CARDINAL
ma-287	189	1	3	CARDINAL
ma-287	189	2	0	CARDINAL
ma-287	189	3	2.1	CARDINAL
ma-287	190	1	1)2	CARDINAL
ma-287	191	1	1)−	CARDINAL
ma-287	193	1	∞∑ k=1	PERSON
ma-287	194	1	2k−1(x2	CARDINAL
ma-287	196	1	− φ	PERSON
ma-287	196	2	2k(x2	CARDINAL
ma-287	197	1	2k−1(x2	CARDINAL
ma-287	198	1	1	CARDINAL
ma-287	198	2	2	CARDINAL
ma-287	198	3	2k(x2	CARDINAL
ma-287	198	4	2	CARDINAL
ma-287	198	5	2k−1(x2	CARDINAL
ma-287	199	1	1)−	ORDINAL
ma-287	199	2	2k(x2−k	CARDINAL
ma-287	199	3	1	CARDINAL
ma-287	200	1	2k−1(x2	CARDINAL
ma-287	203	1	2k−1(x2	CARDINAL
ma-287	203	2	1)22k−1(x2−k	ORDINAL
ma-287	204	1	22(k−1)(x2	CARDINAL
ma-287	205	1	1)3(3 +	DATE
ma-287	206	1	− 1)2	PERSON
ma-287	208	1	1)3(3 +	DATE
ma-287	209	1	4	CARDINAL
ma-287	209	2	1	CARDINAL
ma-287	209	3	2.1	CARDINAL
ma-287	210	1	1− √ log(x	TIME
ma-287	211	1	1)−	CARDINAL
ma-287	213	1	∞∑ k=1	PERSON
ma-287	214	1	2k−1(x2	CARDINAL
ma-287	216	1	− φ	PERSON
ma-287	216	2	2k(x2	CARDINAL
ma-287	218	1	√ 2k−1(x2−(k−1	CARDINAL
ma-287	219	1	1)− √ 2k(x2	DATE
ma-287	219	2	2k−1(x2	CARDINAL
ma-287	221	1	1)−	ORDINAL
ma-287	221	2	2k(x2−k	CARDINAL
ma-287	221	3	1)√	CARDINAL
ma-287	221	4	1	CARDINAL
ma-287	222	1	2k−1(x2	CARDINAL
ma-287	223	1	1	CARDINAL
ma-287	224	1	1)3/2√	CARDINAL
ma-287	225	1	1− √ log(x	TIME
ma-287	226	1	1)3/2√	CARDINAL
ma-287	229	1	j. math	PERSON
ma-287	231	1	10.28924	CARDINAL
ma-287	231	2	1	CARDINAL
ma-287	231	3	2.1	CARDINAL
ma-287	232	1	1	CARDINAL
ma-287	232	2	1)−	CARDINAL
ma-287	234	1	∞∑ k=1	PERSON
ma-287	235	1	2k−1(x2	CARDINAL
ma-287	237	1	− φ	PERSON
ma-287	237	2	2k(x2	CARDINAL
ma-287	238	1	1	CARDINAL
ma-287	239	1	√ 2k−1(x2−(k−1	CARDINAL
ma-287	240	1	1)− √ 2k(x2	DATE
ma-287	242	1	2k−1(x2	CARDINAL
ma-287	244	1	1)−	ORDINAL
ma-287	244	2	2k(x2−k	CARDINAL
ma-287	244	3	1)[√	CARDINAL
ma-287	244	4	1	CARDINAL
ma-287	247	1	2k−1(x2	CARDINAL
ma-287	247	2	1)2[√ 2k−1(x2−k	CARDINAL
ma-287	247	3	1)(1	CARDINAL
ma-287	248	1	1	CARDINAL
ma-287	249	1	√ 22k−1(x2−k	CARDINAL
ma-287	249	2	1)2(1 +	DATE
ma-287	251	1	1	CARDINAL
ma-287	253	1	1	CARDINAL
ma-287	257	1	6	CARDINAL
ma-287	257	2	1	CARDINAL
ma-287	257	3	2.1	CARDINAL
ma-287	261	1	1)−	CARDINAL
ma-287	263	1	∞∑ k=1	PERSON
ma-287	264	1	2k−1(x2	CARDINAL
ma-287	266	1	− φ	PERSON
ma-287	266	2	2k(x2	CARDINAL
ma-287	267	1	2k−1(x2	CARDINAL
ma-287	268	1	2k(x2	CARDINAL
ma-287	269	1	2k−1(x2	CARDINAL
ma-287	269	2	1	CARDINAL
ma-287	270	1	1)(1 + x2−k	DATE
ma-287	271	1	1 + x2	QUANTITY
ma-287	271	2	2	CARDINAL
ma-287	273	1	j. math	PERSON
ma-287	275	1	10.28924	CARDINAL
ma-287	275	2	8hence	DATE
ma-287	278	1	∞∑ k=1	PERSON
ma-287	278	2	1 + x2	QUANTITY
ma-287	278	3	2	CARDINAL
ma-287	279	1	7	CARDINAL
ma-287	279	2	0	CARDINAL
ma-287	279	3	2.1	CARDINAL
ma-287	279	4	sin(x − 1)−	FAC
ma-287	280	1	1)−	CARDINAL
ma-287	282	1	∞∑ k=1	PERSON
ma-287	283	1	2k−1(x2	CARDINAL
ma-287	285	1	− φ	PERSON
ma-287	285	2	2k(x2	CARDINAL
ma-287	286	1	2k−1(x2	CARDINAL
ma-287	287	1	2k(x2	CARDINAL
ma-287	288	1	sin(t)− sin(u	ORG
ma-287	289	1	2	CARDINAL
ma-287	290	1	2	CARDINAL
ma-287	290	2	2k−2(x2	CARDINAL
ma-287	291	1	1)−	ORDINAL
ma-287	291	2	2k−1(x2−k	CARDINAL
ma-287	291	3	1	CARDINAL
ma-287	292	1	2k−2(x2	CARDINAL
ma-287	293	1	1	CARDINAL
ma-287	293	2	2	CARDINAL
ma-287	293	3	2k−2(x2	CARDINAL
ma-287	294	1	2k−2(x2	CARDINAL
ma-287	294	2	1)(3 +	DATE
ma-287	296	1	2 +∞∑ k=1 sin	QUANTITY
ma-287	296	2	2k−2(x2	CARDINAL
ma-287	297	1	2k−2(x2	CARDINAL
ma-287	297	2	1)(3 +	DATE
ma-287	298	1	0	CARDINAL
ma-287	298	2	2.1	CARDINAL
ma-287	300	1	1)−	CARDINAL
ma-287	302	1	∞∑ k=1	PERSON
ma-287	303	1	2k−1(x2	CARDINAL
ma-287	305	1	− φ	PERSON
ma-287	305	2	2k(x2	CARDINAL
ma-287	306	1	2k−1(x2	CARDINAL
ma-287	307	1	2k(x2	CARDINAL
ma-287	308	1	sin[(t−u)/2	PERSON
ma-287	310	1	2k−2(x2	CARDINAL
ma-287	311	1	1)−	ORDINAL
ma-287	311	2	2k−1(x2−k	CARDINAL
ma-287	311	3	1	CARDINAL
ma-287	312	1	2k−2(x2	CARDINAL
ma-287	313	1	1	CARDINAL
ma-287	314	1	2k−2(x2	CARDINAL
ma-287	314	2	2k−2(x2	CARDINAL
ma-287	314	3	1)(3 +	DATE
ma-287	315	1	cos(x	PERSON
ma-287	317	1	+∞∑ k=1 sin	TIME
ma-287	317	2	2k−2(x2	CARDINAL
ma-287	317	3	2k−2(x2	CARDINAL
ma-287	317	4	1)(3 +	DATE
ma-287	319	1	j. math	PERSON
ma-287	321	1	10.28924	CARDINAL
ma-287	321	2	0	CARDINAL
ma-287	321	3	2.1	CARDINAL
ma-287	321	4	1)−	ORDINAL
ma-287	322	1	1)−	CARDINAL
ma-287	324	1	∞∑ k=1	PERSON
ma-287	325	1	2k−1(x2	CARDINAL
ma-287	327	1	− φ	PERSON
ma-287	327	2	2k(x2	CARDINAL
ma-287	328	1	2k−1(x2	CARDINAL
ma-287	328	2	2k(x2	CARDINAL
ma-287	330	1	2	CARDINAL
ma-287	330	2	2k−2(x2	CARDINAL
ma-287	331	1	1)−	ORDINAL
ma-287	331	2	2k−1(x2−k	CARDINAL
ma-287	331	3	1	CARDINAL
ma-287	332	1	2k−2(x2	CARDINAL
ma-287	332	2	1	CARDINAL
ma-287	333	1	2	CARDINAL
ma-287	333	2	2k−2(x2	CARDINAL
ma-287	333	3	2k−2(x2	CARDINAL
ma-287	333	4	1)(3 +	DATE
ma-287	334	1	1)−	ORDINAL
ma-287	335	1	2 +∞∑ k=1	QUANTITY
ma-287	335	2	2k−2(x2	CARDINAL
ma-287	335	3	2k−2(x2	CARDINAL
ma-287	335	4	1)(3 +	DATE
ma-287	336	1	10	CARDINAL
ma-287	336	2	0	CARDINAL
ma-287	336	3	2.1	CARDINAL
ma-287	338	1	1)−	ORDINAL
ma-287	339	1	1)−	CARDINAL
ma-287	341	1	∞∑ k=1	PERSON
ma-287	342	1	2k−1(x2	CARDINAL
ma-287	344	1	− φ	PERSON
ma-287	344	2	2k(x2	CARDINAL
ma-287	345	1	2k−1(x2	CARDINAL
ma-287	346	1	2k(x2	CARDINAL
ma-287	348	1	2	CARDINAL
ma-287	350	1	2	CARDINAL
ma-287	350	2	2k−2(x2	CARDINAL
ma-287	351	1	1)−	ORDINAL
ma-287	351	2	2k−1(x2−k	CARDINAL
ma-287	351	3	1	CARDINAL
ma-287	352	1	2k−2(x2	CARDINAL
ma-287	353	1	1	CARDINAL
ma-287	354	1	2	CARDINAL
ma-287	354	2	2k−2(x2	CARDINAL
ma-287	354	3	2k−2(x2	CARDINAL
ma-287	354	4	1)(3 +	DATE
ma-287	355	1	1)−	ORDINAL
ma-287	356	1	2 +∞∑ k=1	QUANTITY
ma-287	356	2	2k−2(x2	CARDINAL
ma-287	356	3	2k−2(x2	CARDINAL
ma-287	356	4	1)(3 +	DATE
ma-287	359	1	j. math	PERSON
ma-287	361	1	10.28924	CARDINAL
ma-287	361	2	10thus	CARDINAL
ma-287	361	3	2.2	CARDINAL
ma-287	361	4	2.1	CARDINAL
ma-287	361	5	φ	ORG
ma-287	361	6	φ(t	ORG
ma-287	362	1	1/√t	CARDINAL
ma-287	362	2	φ(t	ORG
ma-287	362	3	φ(t	ORG
ma-287	362	4	one	CARDINAL
ma-287	364	1	0	CARDINAL
ma-287	364	2	2.1	CARDINAL
ma-287	364	3	4 as√	QUANTITY
ma-287	364	4	1|	CARDINAL
ma-287	368	1	1|3/2√ 1	DATE
ma-287	370	1	1	CARDINAL
ma-287	370	2	0	CARDINAL
ma-287	370	3	1	CARDINAL
ma-287	370	4	1	CARDINAL
ma-287	370	5	5	CARDINAL
ma-287	371	1	φ	ORG
ma-287	371	2	φ(t	PERSON
ma-287	371	3	∈	ORG
ma-287	371	4	φ	ORG
ma-287	371	5	φ(t	ORG
ma-287	372	1	2.2	CARDINAL
ma-287	373	1	3	CARDINAL
ma-287	374	1	− 1)2	PERSON
ma-287	376	1	1)3(3 +	DATE
ma-287	376	2	1	CARDINAL
ma-287	376	3	1	CARDINAL
ma-287	376	4	ϕ(x	LOC
ma-287	376	5	1	CARDINAL
ma-287	376	6	2	DATE
ma-287	378	1	15	CARDINAL
ma-287	378	2	four	CARDINAL
ma-287	380	1	j. math	PERSON
ma-287	382	1	10.28924	CARDINAL
ma-287	382	2	11	CARDINAL
ma-287	382	3	0	CARDINAL
ma-287	384	1	0	CARDINAL
ma-287	386	1	0 5 10	DATE
ma-287	387	1	3	CARDINAL
ma-287	387	2	0	CARDINAL
ma-287	388	1	1 .0 1	MONEY
ma-287	388	2	2 .0	MONEY
ma-287	388	3	0 5 10 15	DATE
ma-287	388	4	2 5 3	DATE
ma-287	388	5	1	CARDINAL
ma-287	388	6	ϕ(x	LOC
ma-287	388	7	1	CARDINAL
ma-287	388	8	2	DATE
ma-287	390	1	15	CARDINAL
ma-287	390	2	0.5	CARDINAL
ma-287	391	1	1.5	CARDINAL
ma-287	392	1	4	CARDINAL
ma-287	393	1	18	CARDINAL
ma-287	394	1	ϕ(x	LOC
ma-287	394	2	0	CARDINAL
ma-287	394	3	0	CARDINAL
ma-287	394	4	us	GPE
ma-287	394	5	2	CARDINAL
ma-287	394	6	2	CARDINAL
ma-287	394	7	chapter 31	LAW
ma-287	394	8	399	CARDINAL
ma-287	394	9	2.2	CARDINAL
ma-287	395	1	4	CARDINAL
ma-287	395	2	1	CARDINAL
ma-287	398	1	1)3/2√	CARDINAL
ma-287	399	1	2	CARDINAL
ma-287	400	1	9	CARDINAL
ma-287	400	2	2.2	CARDINAL
ma-287	400	3	1)−	ORDINAL
ma-287	401	1	1 2x	CARDINAL
ma-287	401	2	2	CARDINAL
ma-287	402	1	j. math	PERSON
ma-287	404	1	10.28924	CARDINAL
ma-287	404	2	12and	CARDINAL
ma-287	404	3	1)−	ORDINAL
ma-287	405	1	1 2x	CARDINAL
ma-287	405	2	2	CARDINAL
ma-287	405	3	1− x)2k+1	TIME
ma-287	405	4	2k + 1	DATE
ma-287	407	1	10	CARDINAL
ma-287	407	2	1)−	ORDINAL
ma-287	408	1	1 2x	CARDINAL
ma-287	408	2	2 + +	DATE
ma-287	408	3	1− x)2k	LAW
ma-287	408	4	2k	CARDINAL
ma-287	409	1	9	CARDINAL
ma-287	409	2	thetrigonometric	ORG
ma-287	410	1	3	CARDINAL
ma-287	411	1	3.1	CARDINAL
ma-287	412	1	2.2	CARDINAL
ma-287	414	1	3.1	CARDINAL
ma-287	415	1	1	CARDINAL
ma-287	417	1	2	CARDINAL
ma-287	418	1	1	CARDINAL
ma-287	419	1	1	CARDINAL
ma-287	420	1	1	CARDINAL
ma-287	422	1	1	CARDINAL
ma-287	422	2	0	CARDINAL
ma-287	424	1	2	CARDINAL
ma-287	425	1	0	CARDINAL
ma-287	425	2	0	CARDINAL
ma-287	426	1	− 1	EVENT
ma-287	428	1	0	CARDINAL
ma-287	430	1	j. math	PERSON
ma-287	432	1	10.28924	CARDINAL
ma-287	432	2	13	CARDINAL
ma-287	433	1	two	CARDINAL
ma-287	434	1	proof 1	CARDINAL
ma-287	434	2	2.2	CARDINAL
ma-287	435	1	1	CARDINAL
ma-287	435	2	6	CARDINAL
ma-287	435	3	2.2	CARDINAL
ma-287	440	1	∞∑ k=1	PERSON
ma-287	440	2	1 + x2	QUANTITY
ma-287	440	3	2	CARDINAL
ma-287	443	1	2	CARDINAL
ma-287	446	1	2	CARDINAL
ma-287	447	1	proof 2	CARDINAL
ma-287	447	2	15	CARDINAL
ma-287	448	1	1	CARDINAL
ma-287	448	2	1 + √ x	QUANTITY
ma-287	448	3	1	CARDINAL
ma-287	449	1	1 + x2	QUANTITY
ma-287	449	2	1	CARDINAL
ma-287	449	3	1	CARDINAL
ma-287	449	4	1 +	CARDINAL
ma-287	450	1	1	CARDINAL
ma-287	450	2	1	CARDINAL
ma-287	451	1	integer n ≥ 1	ORG
ma-287	451	2	1	CARDINAL
ma-287	452	1	1	CARDINAL
ma-287	452	2	1 +	DATE
ma-287	454	1	1	CARDINAL
ma-287	456	1	1 + x2	QUANTITY
ma-287	458	1	k=1 1	FAC
ma-287	459	1	2	CARDINAL
ma-287	459	2	1	CARDINAL
ma-287	460	1	1	CARDINAL
ma-287	460	2	k=1 1	FAC
ma-287	461	1	2n(x2	CARDINAL
ma-287	465	1	2	CARDINAL
ma-287	467	1	1	CARDINAL
ma-287	468	1	1	CARDINAL
ma-287	468	2	1	CARDINAL
ma-287	469	1	1	CARDINAL
ma-287	469	2	1	CARDINAL
ma-287	469	3	1)/ log(x	TIME
ma-287	470	1	1∏+∞	CARDINAL
ma-287	470	2	1 + x2	QUANTITY
ma-287	476	1	j. math	PERSON
ma-287	478	1	10.28924	CARDINAL
ma-287	478	2	14(3	CARDINAL
ma-287	478	3	1	CARDINAL
ma-287	478	4	1	CARDINAL
ma-287	481	1	2	CARDINAL
ma-287	481	2	2	CARDINAL
ma-287	482	1	4	CARDINAL
ma-287	482	2	0	CARDINAL
ma-287	483	1	− 1	EVENT
ma-287	483	2	1	CARDINAL
ma-287	483	3	1∏+∞	CARDINAL
ma-287	484	1	k=1[(1	GPE
ma-287	485	1	2−kx)/2	CARDINAL
ma-287	486	1	1 + e2	QUANTITY
ma-287	490	1	2	CARDINAL
ma-287	490	2	1	CARDINAL
ma-287	491	1	1	CARDINAL
ma-287	492	1	3	CARDINAL
ma-287	493	1	0	CARDINAL
ma-287	493	2	1	CARDINAL
ma-287	494	1	0	CARDINAL
ma-287	494	2	1	CARDINAL
ma-287	494	3	1	CARDINAL
ma-287	494	4	2−k	CARDINAL
ma-287	495	1	1	CARDINAL
ma-287	496	1	1 x	QUANTITY
ma-287	496	2	1 x	QUANTITY
ma-287	496	3	1	CARDINAL
ma-287	498	1	1	CARDINAL
ma-287	498	2	1 x	QUANTITY
ma-287	500	1	1	CARDINAL
ma-287	502	1	1	CARDINAL
ma-287	504	1	second	ORDINAL
ma-287	504	2	1	CARDINAL
ma-287	504	3	3.1	CARDINAL
ma-287	505	1	0	CARDINAL
ma-287	506	1	1	CARDINAL
ma-287	506	2	0	CARDINAL
ma-287	506	3	1	CARDINAL
ma-287	506	4	1	CARDINAL
ma-287	509	1	2.1	CARDINAL
ma-287	509	2	three	CARDINAL
ma-287	509	3	one	CARDINAL
ma-287	509	4	srinivasa ramanujan	ORG
ma-287	509	5	12	CARDINAL
ma-287	509	6	one	CARDINAL
ma-287	509	7	ludwig seidelin	PERSON
ma-287	510	1	15	CARDINAL
ma-287	510	2	david m. bradley	PERSON
ma-287	511	1	1	CARDINAL
ma-287	512	1	j. math	PERSON
ma-287	514	1	10.28924	CARDINAL
ma-287	515	1	15where	CARDINAL
ma-287	515	2	1	CARDINAL
ma-287	515	3	2	CARDINAL
ma-287	515	4	1	CARDINAL
ma-287	515	5	2	DATE
ma-287	517	1	15	CARDINAL
ma-287	517	2	four	CARDINAL
ma-287	517	3	0.5 ∈	CARDINAL
ma-287	517	4	0	CARDINAL
ma-287	517	5	1	CARDINAL
ma-287	518	1	0	CARDINAL
ma-287	519	1	0	CARDINAL
ma-287	521	1	0 5 10 15	DATE
ma-287	521	2	0 .0 4	MONEY
ma-287	521	3	0	CARDINAL
ma-287	523	1	2	CARDINAL
ma-287	523	2	0	CARDINAL
ma-287	524	1	0	CARDINAL
ma-287	530	1	0 5 10 15	DATE
ma-287	531	1	2	CARDINAL
ma-287	532	1	1 .5	CARDINAL
ma-287	534	1	2	CARDINAL
ma-287	534	2	1	CARDINAL
ma-287	534	3	2	DATE
ma-287	536	1	15	CARDINAL
ma-287	536	2	0.5	CARDINAL
ma-287	537	1	1.5	CARDINAL
ma-287	538	1	4	CARDINAL
ma-287	538	2	18	CARDINAL
ma-287	539	1	0	CARDINAL
ma-287	539	2	3	CARDINAL
ma-287	541	1	0	CARDINAL
ma-287	541	2	2	CARDINAL
ma-287	543	1	1	CARDINAL
ma-287	544	1	1	CARDINAL
ma-287	547	1	j. math	PERSON
ma-287	549	1	10.28924	CARDINAL
ma-287	549	2	16similarly	CARDINAL
ma-287	549	3	0	CARDINAL
ma-287	550	1	1	CARDINAL
ma-287	550	2	1	CARDINAL
ma-287	551	1	2	CARDINAL
ma-287	551	2	0	CARDINAL
ma-287	552	1	1 2	CARDINAL
ma-287	556	1	1	CARDINAL
ma-287	558	1	1	CARDINAL
ma-287	562	1	∞∑ k=0	PERSON
ma-287	565	1	4	CARDINAL
ma-287	565	2	0	CARDINAL
ma-287	567	1	4	CARDINAL
ma-287	573	1	∑+∞ k=1 2	PERSON
ma-287	576	1	4	CARDINAL
ma-287	579	1	j. math	PERSON
ma-287	581	1	10.28924	CARDINAL
ma-287	581	2	173.2	CARDINAL
ma-287	585	1	3.2	CARDINAL
ma-287	586	1	1	CARDINAL
ma-287	587	1	1− √ log(x	TIME
ma-287	591	1	4	CARDINAL
ma-287	591	2	2.2	CARDINAL
ma-287	591	3	1	CARDINAL
ma-287	592	1	1− √	TIME
ma-287	594	1	1)3/2√	CARDINAL
ma-287	596	1	1	CARDINAL
ma-287	596	2	integer k ≥ 1	PERSON
ma-287	596	3	1	CARDINAL
ma-287	597	1	1 + x2	QUANTITY
ma-287	597	2	2 √ 2	DATE
ma-287	598	1	2(k−1)/2(x2−k − 1)3/2 ≥ 0	DATE
ma-287	599	1	1− √ log(x	TIME
ma-287	603	1	2k/2(x2	CARDINAL
ma-287	605	1	2−k	CARDINAL
ma-287	606	1	2k(x2−k	CARDINAL
ma-287	606	2	1)]3/2	CARDINAL
ma-287	606	3	2−k	CARDINAL
ma-287	607	1	2−k	CARDINAL
ma-287	610	1	2.1	CARDINAL
ma-287	611	1	3.3	CARDINAL
ma-287	612	1	2.1	CARDINAL
ma-287	612	2	φ	PERSON
ma-287	612	3	⊆	CARDINAL
ma-287	612	4	1	CARDINAL
ma-287	612	5	2	DATE
ma-287	614	1	φ(x − 1)−	GPE
ma-287	615	1	2	CARDINAL
ma-287	615	2	φ	ORG
ma-287	615	3	⊆	CARDINAL
ma-287	615	4	1	CARDINAL
ma-287	615	5	2	DATE
ma-287	617	1	φ(x − 1)−	GPE
ma-287	619	1	ψ(y	PERSON
ma-287	619	2	1	CARDINAL
ma-287	621	1	j. math	PERSON
ma-287	623	1	10.28924	CARDINAL
ma-287	628	1	1	CARDINAL
ma-287	628	2	0	CARDINAL
ma-287	629	1	1)− 1	DATE
ma-287	629	2	0	CARDINAL
ma-287	631	1	integer k ≥ 1	PERSON
ma-287	631	2	2k−1 ≤ 2k	QUANTITY
ma-287	631	3	ψ(2k	PERSON
ma-287	632	1	two	CARDINAL
ma-287	632	2	•	CARDINAL
ma-287	632	3	φ	ORG
ma-287	635	1	− φ	PERSON
ma-287	635	2	ψ(2k	PERSON
ma-287	636	1	2.1	CARDINAL
ma-287	636	2	⊆	CARDINAL
ma-287	636	3	1	CARDINAL
ma-287	636	4	2	DATE
ma-287	638	1	φ(x − 1)−	GPE
ma-287	640	1	∞∑ k=1	PERSON
ma-287	643	1	φ	ORG
ma-287	643	2	ψ(2k	PERSON
ma-287	648	1	2.1	CARDINAL
ma-287	648	2	⊆	CARDINAL
ma-287	648	3	1	CARDINAL
ma-287	648	4	2	DATE
ma-287	650	1	φ(x − 1)−	GPE
ma-287	652	1	∞∑ k=1	PERSON
ma-287	659	1	1	CARDINAL
ma-287	659	2	3	CARDINAL
ma-287	659	3	2.2	CARDINAL
ma-287	659	4	φ(t	ORG
ma-287	659	5	•	CARDINAL
ma-287	659	6	1	CARDINAL
ma-287	659	7	− 1)2	PERSON
ma-287	662	1	22(k−1)(x2	CARDINAL
ma-287	662	2	1)3(3 +	DATE
ma-287	663	1	•	CARDINAL
ma-287	663	2	0	CARDINAL
ma-287	663	3	1	CARDINAL
ma-287	663	4	− 1)2	PERSON
ma-287	665	1	22(k−1)(x2	CARDINAL
ma-287	665	2	1)3(3 +	DATE
ma-287	666	1	integer k ≥ 1	PERSON
ma-287	666	2	1	CARDINAL
ma-287	667	1	0	CARDINAL
ma-287	667	2	0	CARDINAL
ma-287	667	3	1	CARDINAL
ma-287	670	1	3.4	CARDINAL
ma-287	671	1	0	CARDINAL
ma-287	671	2	log(x) ≤ 2	QUANTITY
ma-287	671	3	1	CARDINAL
ma-287	673	1	three	CARDINAL
ma-287	675	1	j. math	PERSON
ma-287	677	1	10.28924	CARDINAL
ma-287	677	2	19	CARDINAL
ma-287	679	1	0	CARDINAL
ma-287	679	2	2	CARDINAL
ma-287	681	1	first	ORDINAL
ma-287	681	2	2.2	CARDINAL
ma-287	681	3	1	CARDINAL
ma-287	682	1	φ(t	ORG
ma-287	682	2	1− log(x	TIME
ma-287	683	1	2k−1(x2	CARDINAL
ma-287	683	2	− 1)2	PERSON
ma-287	684	1	1−	CARDINAL
ma-287	685	1	1)2	CARDINAL
ma-287	685	2	2	CARDINAL
ma-287	685	3	1	CARDINAL
ma-287	686	1	proof 3	CARDINAL
ma-287	687	1	3.3	CARDINAL
ma-287	687	2	ψ(y	PERSON
ma-287	687	3	1	CARDINAL
ma-287	688	1	0	CARDINAL
ma-287	688	2	1	CARDINAL
ma-287	691	1	17],and	ORDINAL
ma-287	692	1	3.5	CARDINAL
ma-287	693	1	1	CARDINAL
ma-287	694	1	1	CARDINAL
ma-287	694	2	2 1	DATE
ma-287	696	1	3	CARDINAL
ma-287	697	1	0	CARDINAL
ma-287	697	2	1	CARDINAL
ma-287	698	1	integer k ≥ 1	PERSON
ma-287	698	2	1	CARDINAL
ma-287	702	1	2 1 + x2 −k ]	DATE
ma-287	702	2	2 1 + x2	QUANTITY
ma-287	702	3	1	CARDINAL
ma-287	703	1	2 1 + x2 −k	DATE
ma-287	704	1	1. •	CARDINAL
ma-287	704	2	0	CARDINAL
ma-287	704	3	1	CARDINAL
ma-287	704	4	integer k ≥ 1	PERSON
ma-287	704	5	≤ 1	DATE
ma-287	708	1	j. math	PERSON
ma-287	710	1	10.28924	CARDINAL
ma-287	710	2	20this	CARDINAL
ma-287	711	1	0	CARDINAL
ma-287	712	1	3	CARDINAL
ma-287	713	1	3.6	CARDINAL
ma-287	714	1	1	CARDINAL
ma-287	716	1	2 1 +	CARDINAL
ma-287	719	1	4	CARDINAL
ma-287	720	1	0	CARDINAL
ma-287	720	2	integer k ≥ 1	PERSON
ma-287	720	3	1	CARDINAL
ma-287	721	1	1	CARDINAL
ma-287	722	1	2 1 +	CARDINAL
ma-287	722	2	2 1 +	CARDINAL
ma-287	723	1	2 1 +	CARDINAL
ma-287	724	1	1	CARDINAL
ma-287	725	1	integer k ≥ 1	PERSON
ma-287	725	2	1	CARDINAL
ma-287	728	1	4	CARDINAL
ma-287	730	1	4	CARDINAL
ma-287	732	1	first	ORDINAL
ma-287	732	2	srinivasa ramanujan	ORG
ma-287	732	3	2	CARDINAL
ma-287	732	4	5	CARDINAL
ma-287	733	1	1	CARDINAL
ma-287	735	1	one	CARDINAL
ma-287	738	1	j. math	PERSON
ma-287	740	1	10.28924	CARDINAL
ma-287	741	1	1	CARDINAL
ma-287	741	2	m. abramowitz	PERSON
ma-287	741	3	i.a.	GPE
ma-287	742	1	27.1	CARDINAL
ma-287	742	2	9th	ORDINAL
ma-287	743	1	new york	GPE
ma-287	743	2	dover	GPE
ma-287	743	3	999-1000	CARDINAL
ma-287	743	4	b.c. berndt	ORG
ma-287	743	5	ramanujan	GPE
ma-287	743	6	new york	GPE
ma-287	743	7	1994	DATE
ma-287	744	1	978	CARDINAL
ma-287	744	2	l. bougoffa	PERSON
ma-287	744	3	p. krasopoulos	PERSON
ma-287	744	4	journal of inequalitiesand	ORG
ma-287	744	5	12	CARDINAL
ma-287	744	6	24-32	DATE
ma-287	744	7	2021.[4	DATE
ma-287	744	8	bradley	PERSON
ma-287	745	1	mag	PERSON
ma-287	746	1	90 (	PERCENT
ma-287	746	2	2017	CARDINAL
ma-287	746	3	353	CARDINAL
ma-287	747	1	bradley	PERSON
ma-287	747	2	ramanujan	GPE
ma-287	747	3	j. 47	PERSON
ma-287	747	4	2018),253-265	DATE
ma-287	748	1	f. burk	PERSON
ma-287	749	1	mon.	DATE
ma-287	750	1	1987	DATE
ma-287	750	2	527	CARDINAL
ma-287	751	1	https	PERSON
ma-287	751	2	y.j.	GPE
ma-287	752	1	j. math	PERSON
ma-287	755	1	8	CARDINAL
ma-287	755	2	2020	DATE
ma-287	755	3	140	CARDINAL
ma-287	756	1	gradshteyn	PERSON
ma-287	756	2	i.m.	PERSON
ma-287	756	3	ryzhik	NORP
ma-287	756	4	russian	NORP
ma-287	756	5	eighth	ORDINAL
ma-287	756	6	seventh	ORDINAL
ma-287	756	7	d.	NORP
ma-287	756	8	moll	PERSON
ma-287	757	1	amsterdam	GPE
ma-287	757	2	2015.[9	CARDINAL
ma-287	757	3	g. jameson	PERSON
ma-287	757	4	p.r.	PERSON
ma-287	757	5	mercer	PERSON
ma-287	758	1	mon	PERSON
ma-287	759	1	126	CARDINAL
ma-287	759	2	2019	CARDINAL
ma-287	759	3	641	CARDINAL
ma-287	760	1	m. kostić	PERSON
ma-287	760	2	2020	DATE
ma-287	760	3	291	CARDINAL
ma-287	761	1	e.r.	GPE
ma-287	763	1	64	CARDINAL
ma-287	763	2	1980	DATE
ma-287	763	3	55	CARDINAL
ma-287	764	1	s. ramanujan	PERSON
ma-287	764	2	2	CARDINAL
ma-287	764	3	tata institute of fundamental research	ORG
ma-287	764	4	bombay	GPE
ma-287	765	1	j. 53	PERSON
ma-287	765	2	2022	CARDINAL
ma-287	765	3	1	CARDINAL
ma-287	765	4	https	PERSON
ma-287	765	5	w. rudin	PERSON
ma-287	765	6	mcgraw-hill	ORG
ma-287	765	7	new york	GPE
ma-287	765	8	1986.[15	CARDINAL
ma-287	765	9	l. seidel	PERSON
ma-287	765	10	ueber eine darstellung des kreisbogens	PERSON
ma-287	765	11	durchunendliche producte	PERSON
ma-287	765	12	j. reine angew	PERSON
ma-287	767	1	73 (1871	DATE
ma-287	767	2	273	CARDINAL
ma-287	767	3	n. thomas	PERSON
ma-287	767	4	a. chandran	PERSON
ma-287	767	5	k. namboothiri	PERSON
ma-287	768	1	j. 10	PERSON
ma-287	768	2	2021	CARDINAL
ma-287	768	3	2483-2489	CARDINAL
ma-287	769	1	a. vallejo	PERSON
ma-287	769	2	2023	CARDINAL
ma-287	770	1	arxiv	ORG
ma-287	770	2	1	CARDINAL
ma-287	775	1	2	CARDINAL
ma-287	775	2	2.1	CARDINAL
ma-287	776	1	2.2	CARDINAL
ma-287	776	2	3	CARDINAL
ma-287	776	3	3.1	CARDINAL
ma-287	776	4	3.2	CARDINAL
ma-287	776	5	4	CARDINAL