Special values of the Lerch zeta function and the evaluation of certain integrals
Authors: Williams Kenneth S. , Zhang Nan Yue 
Published: June 2010

The Lerch zeta function

Φ
(
x
,
a
,
s
)

\Phi (x,a,s)

is defined by the series
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi left-parenthesis x comma a comma s right-parenthesis equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts StartFraction e Superscript 2 n pi i x Baseline Over left-parenthesis n plus a right-parenthesis Superscript s Baseline EndFraction comma">

Φ
(
x
,
a
,
s
)
=

∑

n
=
0

∞

e

2
n
π
i
x

(
n
+
a
)

s

,

\Phi (x,a,s) = \sum \limits _{n = 0}^\infty {\frac {{{e^{2n\pi ix}}}} {{{{(n + a)}^s}}}} ,

\]
where

x
x

is real,

0
>
a
⩽
1

0 > a \leqslant 1

, and

σ
=
Re
⁡
(
s
)
>
1

\sigma = \operatorname {Re} (s) > 1

if

x
x

is an integer and

σ
>
0

\sigma > 0

otherwise. In this paper we study the function

J

(

s
,
a

)

=
Φ
(

1
2

,
a
,
s
)

J\left ( {s,a} \right ) = \Phi (\tfrac {1} {2},a,s)

. We use its integral representation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J left-parenthesis s comma a right-parenthesis equals StartFraction a Superscript negative s Baseline Over 2 EndFraction plus 2 integral Subscript 0 Superscript normal infinity Baseline left-parenthesis a squared plus y squared right-parenthesis Superscript negative s slash 2 Baseline sine left-parenthesis s tangent Superscript negative 1 Baseline StartFraction y Over a EndFraction right-parenthesis StartFraction e Superscript pi y Baseline d y Over e Superscript 2 pi y Baseline minus 1 EndFraction">

J

(

s
,
a

)

=

a

−
s

2

+
2

∫
0
∞

(

a
2

+

y
2

)

−
s

/

2

sin
⁡

(

s

tan

−
1

y
a

)

e

π
y

d
y

e

2
π
y

−
1

J\left ( {s,a} \right ) = \frac {{{a^{ - s}}}} {2} + 2\int _0^\infty {{{({a^2} + {y^2})}^{ - s/2}}\sin \left ( {s\,{{\tan }^{ - 1}}\frac {y} {a}} \right )} \frac {{{e^{\pi y}}dy}} {{{e^{2\pi y}} - 1}}

\]
to obtain the values of certain definite integrals; for example, we show that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row integral Subscript 0 Superscript normal infinity Baseline StartFraction hyperbolic cosine x log x Over hyperbolic cosine 2 x minus cosine 2 pi a EndFraction d x 2nd Row equals StartFraction pi Over 2 sine pi a EndFraction left-brace log StartFraction normal upper Gamma left-parenthesis left-parenthesis 1 plus a right-parenthesis slash 2 right-parenthesis Over normal upper Gamma left-parenthesis a slash 2 right-parenthesis EndFraction plus one half log left-parenthesis 2 pi cotangent StartFraction pi a Over 2 EndFraction right-parenthesis right-brace comma 0 greater-than a greater-than 1 period EndLayout">

∫
0
∞

cosh
⁡
x
log
⁡
x

cosh
⁡
2
x
−
cos
⁡
2
π
a

d
x

=

π

2
sin
⁡
π
a

{

log
⁡

Γ
(
(
1
+
a
)

/

2
)

Γ
(
a

/

2
)

+

1
2

log
⁡

(

2
π
cot
⁡

π
a

2

)

}

,

0
>
a
>
1.

\begin {gathered} \int _0^\infty {\frac {{\cosh x\log x}} {{\cosh 2x - \cos 2\pi a}}} dx \hfill \\ \qquad = \frac {\pi } {{2\sin \pi a}}\left \{ {\log \frac {{\Gamma ((1 + a)/2)}} {{\Gamma (a/2)}} + \frac {1} {2}\log \left ( {2\pi \cot \frac {{\pi a}} {2}} \right )} \right \},\qquad 0 > a > 1. \hfill \\ \end {gathered}

\]