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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 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
nζ
∞
==∑ , (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
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
( ) ( )( )
( 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.
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