14_xxx_rusev.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 6, 2010, 1113-1117

ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE ON COMPLEX ANALYSIS: THEORY AND APPLICATIONS

DEDICATED TO PROFESSOR HARI M. SRIVASTAVA,

ON THE OCCASION OF HIS 70TH
BIRTHDAY

Hankel’s Transform and Riemann’s Hypothesis

Peter Rusev

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str.,

bL. 8, 1113 Sofia, Bulgaria

Abstract. A necessary and sufficient condition for validity of Rieman’s hypothesis is given in terms of

the growth of Hankel’s transform of a function closely related to the classical ζ-function.

2000 Mathematics Subject Classifications: 11M26, 33C45, 42A38

Key Words and Phrases: Laguerre polynomials, Hankel transform, Riemann’s hypothesis

1. Expansion of Holomorphic Functions in Series of the Polynomials

{L(α)
n
(z2)}∞

n=0

It is well-known that the region of convergence of a series in Laguerre’s polynomial

{L(α)n (z)}∞n=0,α > −1 is, in general, the interior ∆(λ0) of the parabola with equation

ℜ(−z)1/2 = λ0, 0 < λ0 ≤ ∞, [11, 9.2., (5)]. A corollary of this fact is that the region of

convergence of a series of the kind

∞
∑

n=0

an L(α)n (z
2), α > −1 (1)

is a strip S(λ0) defined by the inequality |ℑz| < λ0 [5, 1. Introduction].

Denote by P (α)(λ0), 0 < λ0 ≤ ∞,α > −1 the C-vector space of the even complex func-

tions holomorphic in the strip S(λ0) and having there a representation by a series of the kind

(1).

Email address: pkrusev�math.bas.bg
http://www.ejpam.com 1113 c© 2010 EJPAM All rights reserved.



P. Rusev / Eur. J. Pure Appl. Math, 3 (2010), 1113-1117 1114

The space P (0)(λ0) or, more precisely, the growth of the functions in it is completely

characterized first by H. Pollard [5, Theorem A] by means of the function

η(λ; x , y) = exp{x2/2− |x |(λ2 − y2)1/2}, 0 ≤ λ < ∞, x + i y ∈ S(λ), S(0) := R actually

introduced in E. Hille’s paper [4]. In fact, Pollard has proved that:

Theorem 1. A complex function f , holomorphic in the strip S(λ0), 0 < λ0 ≤∞, is in the space

P (0)(λ0) iff for each λ ∈ [0,λ0) and z = x + i y ∈ S(λ),

| f (z)| = | f (x + i y)| = O(η(λ; x , y)). (2)

Pollard’s theorem has been generalized by O. Százs and N. Yeardley [10, Theorem A], who

proved that if α > −1, then a function f , holomorphic in the strip S(λ0), 0 < λ0 ≤ ∞, is in

the class P (α)(λ0) iff it satisfies (2).

2. Hankel’s Transform and Series Representation by Laguerre Polynomials

Another approach to the series representation of the kind (1) is based on the integral

representation of Laguerre’s polynomials by means of Bessel’s functions of first kind [1, 10.12.,

(21)] as well as on the class G(λ),−∞ < λ ≤ ∞ of entire functions F of exponential type

introduced in [6, Definition 1.] by the requirement

lim sup
|w|→∞

(2
p

|w|)−1(log |F(w)| − |w|)≤ −λ.

The corresponding proposition is announced in [6, Theorem 1.] and says that:

Proposition. Let 0 < λ0 ≤∞ and α > −1. A complex function f analytic in the region ∆(λ0)

can be represented in this region as a series of Laguerre polynomials {L(α)n (z)}∞n=0 if and only if

the following representation holds in the region ∆(λ0) \ (−λ0, 0]:

f (z) = z−α/2ez

∫ ∞

0

tα/2et F(t)Jα(2
p

zt) d t

where Jα is the Bessel function of the first kind of order α and the function F ∈ A(λ0).

Remark. A proof can be found in [7] as well as in [8, Chapter VI, 1].

Let f be an even complex function holomorphic in the strip S(λ0), 0< λ0 ≤∞. Then, the

function f (
p

z) is holomorphic in the region ∆(λ0) \ (−λ2
0, 0]. Since f is even,

limz→x ,ℑz>0 f (
p

z) = limz→x ,ℑz<0 f (
p

z) for each x ∈ (−λ2
0, 0), i.e. f has a continuous exten-

sion in the region ∆(λ0). In fact, f is holomorphic there and this can be proved e.g. by an

usual use of Morera’s theorem [9, (8.1)].

Suppose now that 0< λ0 ≤∞,α > −1 and that an even complex function f , holomorphic

in the strip S(λ0), has representation by the series (1) in this strip. Then, the function f (
p

z)



P. Rusev / Eur. J. Pure Appl. Math, 3 (2010), 1113-1117 1115

admits representation in the region ∆(λ0) by series in the polynomials {L(α)n (z)}∞n=0 and,

hence, the representation

zα/2 exp(−z) f (
p

z) =

∫ ∞

0

tα/2 exp(−t)F(t)Jα(2
p

zt) d t

holds in the region ∆(λ0) \ (−λ2
0, 0]. Replacing z by z2, we obtain that the representation

zα exp(−z2) f (z) =

∫ ∞

0

tα/2 exp(−t)F(t)Jα(2z
p

t) d t

holds in the half-strip S+(λ0) := { z ∈ S(λ0) : ℜz > 0}. The converse is also true, i.e. if the

above representation holds for an even function f holomorphic in the strip S(λ0), then it has

a representation there by a series in the polynomials {L(α)n (z
2)}∞n=0. Further, replacing z by

z/
p

2 and changing t by t2/2,we come to the following assertion:

Assertion. An even complex function f , holomorphic in the strip S(λ0), 0 < λ0 ≤ ∞ is in the

space P (α)(λ0),α > −1 iff the representation

zα+1/2 exp(−z2) f (z/
p

2)

=

∫ ∞

0

tα+1/2 exp(−t2) f (t2/2)(zt)1/2Jα(zt) d t

holds in the half-strip S+(λ0) with a function F ∈ G(λ0).

3. The Main Results

Since the Riemann function ζ(s), s = σ+i t does not vanish on the closed half-plane σ ≥ 1,

there exists a region B containing this half-plane and such that ζ(s) 6= 0 for s ∈ B. Hence, the

function

Φ(s) = − ζ
′(s)

sζ(s)
− 1

s− 1

is holomorphic in the region B. Moreover, the integral representation

Φ(s) =

∫ ∞

1

ψ(x)− x

x s+1
d x (3)

holds on the closed half-plane σ ≥ 1, where ψ is one of the Chebisheff functions [3, Chap-

terXI, Section 3]. A corollary of (3) is that the function Φ is bounded in this half-plane. Indeed,

since ψ(x)− x = O(x exp(−c(log x)1/2)), c > 0 as x →∞ [3, Section 18, (1)], we have that

for σ ≥ 1 and −∞< t <∞,

|Φ(s)| ≤
∫ ∞

1

|ψ(x)− x |
xσ+1

d x = O

�
∫ ∞

1

x−1 exp(−c(log x)1/2) d x

�



P. Rusev / Eur. J. Pure Appl. Math, 3 (2010), 1113-1117 1116

= O

�
∫ ∞

0

exp(−cx1/2) d x

�

= O(1).

Suppose now that the function ζ has no zeros in the half-plane σ > θ , 1/2 ≤ θ < 1. Then,

ψ(x) = x +O(xθ log2 x) as x →∞ [3, Section 18], i.e. whatever ǫ > 0 may be,

ψ(x) = x + O(xθ+ǫ) as x → ∞. Then, the integral in (3) is uniformly convergent on each

closed half-plane σ ≥ θ + ǫ. That means the function Φ(s) is analytically continuable in each

half-plane σ > θ + ǫ and, moreover, it is bounded when σ ≥ θ + ǫ. Since ǫ > 0 is arbitrary,

it follows that, in fact, Φ is holomortphic in the half-plane σ > θ and bounded in each half-

plane σ ≥ θ + ǫ,ǫ > 0. Hence, the function Φ̃(s) = Φ(s)+Φ(2− s) is holomorphic in the strip

θ < σ < 2− θ and is bounded in each closed strip θ + ǫ ≤ σ ≤ 2− θ − ǫ provided

0< ǫ < 1−θ . Therefore, the even function Φ∗(z) = Φ̃(1+ iz) = Φ(1+ iz)+Φ(1− iz) is holo-

morphic in the strip S(1− θ) and, moreover, it is bounded on each closed strip S(1− θ − ǫ)
with ǫ ∈ (0,1− θ). That means the function Φ∗ is in the space P (α)(1− θ) for each α > −1,

i.e. there is a function F ∈ G(1− θ) such that

zα+1/2 exp(−z2/2)Φ∗(z/
p

2) =

∫ ∞

0

tα+1/2 exp(−t2/2)F(t2/2)(zt)1/2Jα(zt) d t (4)

for z ∈ S+(1− θ) and, hence,

tα+1/2 exp(−t2/2)F(t2/2) =

∫ ∞

0

xα+1/2 exp(−x2/2)Φ∗(x/
p

2)(t x)1/2Jα(t x) d x .

We have just proved that if ζ(s) 6= 0 for σ > θ , 1/2 ≤ θ < 1, then the Hankel transform with

kernel w1/2Jα(w),α > −1 of the function

xα+1/2 exp(−x2/2)Φ∗(x/
p

2), 0< x <∞ (5)

is of the form

tα+1/2 exp(−t2/2)F(t2/2), 0< t <∞ (6)

with function F ∈ G(1− θ).
The converse is also true. Indeed, suppose the Hankel transform with kernel

w1/2Jα(w), α > −1 of the function (5) is of the form (6) with function F in the class G(1−θ),
1/2 ≤ θ < 1, i.e. (4) holds for z = x ∈ (0,∞). By means of the asymptotic formula [1,

7.13, (3)] for the function Jα(z) it can be proved that whatever ǫ ∈ (0,1− θ) may be, the

integral in (4) is uniformly convergent in the strip S(
p

2(1−θ−ǫ)). That means the function

Φ∗(x/
p

2), 0 < x <∞ has a holomorphic extension in the half-strip S+(
p

2(1− θ)), i.e. the

function Φ∗(x) has a holomorphic extension in the strip S(1− θ). Therefore, the function

Φ(s) is analytically continuable in the half-plane σ > θ and, hence, ζ(s) 6= 0 in this half-plane.

Thus we have proved that:

Theorem 2. A necessary and sufficient condition the function ζ(s) to have no zeros in the half-

plane σ > θ , 1/2 ≤ θ < 1 is the Hankel transform with kernel w1/2Jα(w) of the function (5) to

be of the form (6).



REFERENCES 1117

A corollary of the above assertions is the following criterion:

Corollary. Riemann’s hypothesis is true iff the Hankel transform with kernel w1/2Jα(w) of the

function (5) is of the form (6) with a function F ∈ G(1/2).

ACKNOWLEDGEMENTS This paper is partially supported by Project D ID O2/25/2009 "In-

tegral Transform Methods, Special Functions and Applications", National Science Fund, Min-

istry of Education, Youth and Science, Bulgaria.

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