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On the functional equation for the Hurwitz zeta-function Yang Ming-shun (Department of Mathematics, Weinan Teacher’s College, Weinan 714000, China) e-mail [email protected] Abstract: In this note, we shall derive the functional equation for the Hurwitz zeta-function ( ) , s u ζ from that for the Riemann zeta-function () s ζ , on using an integral expression for ( ) , s u ζ which in turn depends on the functional equation for () s ζ . Key word: Riemann zeta-function, Hurwitz zeta-function, functional equation,integral representation. §1 introduction and the result Throughout this note we let it s + = σ , β α i z + = signify the complex variables. The Riemann zeta-function () s ζ is defined for 1 > σ by the absolutely convergent Dirichlet series Foundation project: Supported by the Natural Science Foundation of China (10671155) and Natural Science Foundation of Weinan teachers’ college ( 09YKS001) Biography: Yang Mingshun (1964-), male , native of Weinan, Shannxi, an associate professor of Weinan Teacher’s college ,engage in number theory () 1 1 s n s n ζ = = , (1) where n s s e n log = , n log taking the principal value, is meromorphically continued over the whole s-plane and satisfies the functional equation. 2 2 s s π Γ () s ζ = 1 2 1 2 s s π Γ ( ) 1 s ζ , (2) where () s Γ signifies the gamma function. The Hurwitz zeta-function ( ) u s, ς is defined for 1 > σ , 1 0 < u by ( ) , s u ζ = = + 0 ) ( 1 n s u n ( ( ) ,1 s ζ = () s ζ ) (3) and is continued meromorphically over the whole plane and satisfies the functional equation ( ) () ( ) 1 , 2 s s su ζ π Γ = ( 2 is e π ( ) ( ) 2 1 is s s l u e l u π + ) (4) for 1 > σ and 1 0 < < u , where ( ) 2 1 inu s s n e l u n π = = (5) 2009 International Conference on Artificial Intelligence and Computational Intelligence 978-0-7695-3816-7/09 $26.00 © 2009 IEEE DOI 10.1109/AICI.2009.73 595

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Page 1: [IEEE 2009 International Conference on Artificial Intelligence and Computational Intelligence - Shanghai, China (2009.11.7-2009.11.8)] 2009 International Conference on Artificial Intelligence

On the functional equation for the Hurwitz zeta-function

Yang Ming-shun

(Department of Mathematics, Weinan Teacher’s College, Weinan 714000, China)

e-mail [email protected]

Abstract: In this note, we shall derive the functional equation for the

Hurwitz zeta-function ( ),s uζ from

that for the Riemann zeta-function

( )sζ , on using an integral expression

for ( ),s uζ which in turn depends on

the functional equation for ( )sζ .

Key word: Riemann zeta-function, Hurwitz zeta-function, functional equation,integral representation.

§1 introduction and the result Throughout this note we let

its += σ , βα iz += signify the

complex variables.

The Riemann zeta-function ( )sζ

is defined for 1>σ by the absolutely convergent Dirichlet series

Foundation project: Supported by the Natural

Science Foundation of China (10671155) and Natural

Science Foundation of Weinan teachers’ college

( 09YKS001)

Biography: Yang Mingshun (1964-), male ,

native of Weinan, Shannxi, an associate professor of

Weinan Teacher’s college ,engage in number theory

( )1

1s

ns

==∑ , (1)

where nss en log−− = , nlog taking

the principal value, is meromorphically continued over the whole s-plane and satisfies the functional equation.

2

2

s sπ− ⎛ ⎞Γ ⎜ ⎟

⎝ ⎠( )sζ =

12 1

2

s sπ−− −⎛ ⎞Γ ⎜ ⎟

⎝ ⎠( )1 sζ − , (2)

where ( )sΓ signifies the gamma

function.

The Hurwitz zeta-function ( )us,ς is

defined for 1>σ , 10 ≤< u by

( ),s uζ = ∑∞

= +0 )(1

nsun

( ( ),1sζ = ( )sζ ) (3)

and is continued meromorphically over the whole plane and satisfies the functional equation

( ) ( )( )

1 ,2 s

ss uζ

πΓ

− =

( 2i s

eπ− ( ) ( )2 1

is

s sl u e l uπ

+ − ) (4)

for 1>σ and 10 << u , where

( )2

1

inu

s sn

el un

π∞

=

=∑ (5)

2009 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-3816-7/09 $26.00 © 2009 IEEE

DOI 10.1109/AICI.2009.73

595

Page 2: [IEEE 2009 International Conference on Artificial Intelligence and Computational Intelligence - Shanghai, China (2009.11.7-2009.11.8)] 2009 International Conference on Artificial Intelligence

absolutely convergent for 1>σ , is the polylogarithm function

The limiting case u −→ 1 of (4) leads to the asymmetric form of the

functional equation for ( )sζ

( )1 sζ − = ( )sss Γ−− π12 cos2sπ ( )sζ (6)

which is equivalent to (2), whence (4)⇒ (2).

Recently, there have appeared two proofs of the eguivalence (2) ⇔ (4), i.e. (2) implies (4). The proof in [1] appeals to the general theory of modular relations and makes use of hypergeometric functions while the proof in [3] appeals to partial fraction expansion for the cotangent function, which is an aspect of the modular relation. For a detailed account of this topic , see [5]

In this short note, we shall give a simple proof of the implication (2)⇒ (4), using a result of Mikolás [4], which in turn depends on (2) and is stated as

Lemma ([4, Satz 2]). For 121 ≤< σ and

211 <<− ασ , we have the integral

representation

( )( )

( )11 , ,2

zs u u F s z dz

i αζ

π−

− = ∫ (7)

where the integral is over the vertical line itz += α , ∞<<∞− t and

( ) ( )( )

( ) ( ) ( ), 2 2 cos22

zs

sF s z s z zππ

π−Γ

= − Γ

( )s zζ + (8)

The proof depends on the Cauchy residue theoem and in computing the residues the expansion

( ) ( ) ( ) ( )0

, 1s s kk

ks u u s k u sζ ζ

∞− −

== + + ≠∑

(9) is used. We also need the formula

( )( )1 , 0, Re 0

2z XX z dz e X

i αα

π− −Γ = > ≥∫

(10) which can be obtained as the Mellin inversion of the formula

20

1zi

itz edtetπ

−−∞ − =∫ ( )zΓ

(cf .e. g. [2, p. 2] ) Theorem: The functional

equation (2) for ( )sζ implies the

functional equation (4) for ( ),s uζ

§2 Proof of Theorem

Substituting the series expansion

(1) for ( )s zζ + in (6) and changing

the order of summation and integration (permissible because of absolute

convergence), we obtain for 121 << σ ,

10 << u ,

( ) ( )( )

( )( )

( ) ( )1

1 11 , 2 cos2 22

z

s sn

ss u nu z s z dz

n i α

πζ πππ

∞ −

=

Γ− = Γ −∑ ∫

Substituting

( ) ( ) ( )2 21cos

2 2

i is z s zs z e e

π ππ − − −⎛ ⎞− = +⎜ ⎟

⎝ ⎠, we

transform the above expression further:

( ) ( )( )

( ( )( )

( )2

1

11 , 22

i s z

s sn

ss u e inu z dz

n

π

αζ π

π

∞ −

=

Γ− = Γ∑ ∫

( )( )

( ) )2 2i s z

e inu z dzπ

απ

−−+ − Γ∫

which becomes , by (10)

596

Page 3: [IEEE 2009 International Conference on Artificial Intelligence and Computational Intelligence - Shanghai, China (2009.11.7-2009.11.8)] 2009 International Conference on Artificial Intelligence

( ) ( )( )

( 22

1

11 ,2

i si nu

s sn

ss u e e

n

ππζ

π

∞−

=

Γ− = ∑

22i s

inue eπ

π−+ )=

( )( )

(2 (1 )

2

12

is in u

s sn

s een

π π

π

−∞

=

Γ∑ + )

22

1

is inu

sn

een

π π∞−

=∑

which is (4), on remarking that the series

(5) for ( )sl u is uniformly convergent

for 0>σ , (cf. e.g.[5, p. 205]). Hence by analytic continuation, (4) is valid for

10 << σ . This completes the proof. Remark. Mikol á s could have

easily deduced our theorem if he had known that the implication (2)⇒ (4) is possible which does not look possible at a first glance. Indeed, he states two applications of Lemma, but no mention is made on the above consequence.

References [1] R. Balasalramanian, L. Ding, S.

Kanemitsu and Y. Tanigawa, On the

partial fraction expansion for the

cotangent function, Proc. Chandigarh

Conf. (2005), to appear

[2] M. A. Chaudhry and S. M. Zubair, On a class of incomplete gamma functions

with applications, Chapman Hall /CRC,

New York etc.2002

[3] S. Kanemitsu, H. Tsukada and Y. Tanigawa, Some number theoretic applications of a general modular relation, Int. J. Number Theory 2 (2007), 1-17

[4] M. Mikol á s, Mellinsche Transformation und Orthogonalitat bei

( ),s uζ ; Verallgemeinerung der

Riemanschen Funktionalgleichung von

( )sζ , Acta Sci Math. (Szeged) 17(1965),

143-164 [5] S. Kanemitsu and H. Tsukada, Vistas

of special functions, World Scientific, Singapore etc. 2007, to appear.

597