In mathematics, the Lerch zeta-function, sometimes called the Hurwitz–Lerch zeta-function, is a special function that generalizes the Hurwitz zeta-function and the polylogarithm. It is named after the Czech mathematician Mathias Lerch [1].
The Lerch zeta-function is given by
L
(
λ
,
α
,
s
)
=
∑
n
=
0
∞
exp
(
2
π
i
λ
n
)
(
n
+
α
)
s
.
A related function, the Lerch transcendent, is given by
Φ
(
z
,
s
,
α
)
=
∑
n
=
0
∞
z
n
(
n
+
α
)
s
.
The two are related, as
Φ
(
exp
(
2
π
i
λ
)
,
s
,
α
)
=
L
(
λ
,
α
,
s
)
.
An integral representation is given by
Φ
(
z
,
s
,
a
)
=
1
Γ
(
s
)
∫
0
∞
t
s
−
1
e
−
a
t
1
−
z
e
−
t
d
t
for
ℜ
(
a
)
>
0
∧
ℜ
(
s
)
>
0
∧
z
<
1
∨
ℜ
(
a
)
>
0
∧
ℜ
(
s
)
>
1
∧
z
=
1.
A contour integral representation is given by
Φ
(
z
,
s
,
a
)
=
−
Γ
(
1
−
s
)
2
π
i
∫
0
(
+
∞
)
(
−
t
)
s
−
1
e
−
a
t
1
−
z
e
−
t
d
t
for
ℜ
(
a
)
>
0
∧
ℜ
(
s
)
<
0
∧
z
<
1
where the contour must not enclose any of the points
t
=
log
(
z
)
+
2
k
π
i
,
k
∈
Z
.
A Hermite-like integral representation is given by
Φ
(
z
,
s
,
a
)
=
1
2
a
s
+
∫
0
∞
z
t
(
a
+
t
)
s
d
t
+
2
a
s
−
1
∫
0
∞
sin
(
s
arctan
(
t
)
−
t
a
log
(
z
)
)
(
1
+
t
2
)
s
/
2
(
e
2
π
a
t
−
1
)
d
t
for
ℜ
(
a
)
>
0
∧
|
z
|
<
1
and
Φ
(
z
,
s
,
a
)
=
1
2
a
s
+
log
s
−
1
(
1
/
z
)
z
a
Γ
(
1
−
s
,
a
log
(
1
/
z
)
)
+
2
a
s
−
1
∫
0
∞
sin
(
s
arctan
(
t
)
−
t
a
log
(
z
)
)
(
1
+
t
2
)
s
/
2
(
e
2
π
a
t
−
1
)
d
t
for
ℜ
(
a
)
>
0.
The Hurwitz zeta-function is a special case, given by
ζ
(
s
,
α
)
=
L
(
0
,
α
,
s
)
=
Φ
(
1
,
s
,
α
)
.
The polylogarithm is a special case of the Lerch Zeta, given by
Li
s
(
z
)
=
z
Φ
(
z
,
s
,
1
)
.
The Legendre chi function is a special case, given by
χ
n
(
z
)
=
2
−
n
z
Φ
(
z
2
,
n
,
1
/
2
)
.
The Riemann zeta-function is given by
ζ
(
s
)
=
Φ
(
1
,
s
,
1
)
.
The Dirichlet eta-function is given by
η
(
s
)
=
Φ
(
−
1
,
s
,
1
)
.
For λ rational, the summand is a root of unity, and thus
L
(
λ
,
α
,
s
)
may be expressed as a finite sum over the Hurwitz zeta-function.
Various identities include:
Φ
(
z
,
s
,
a
)
=
z
n
Φ
(
z
,
s
,
a
+
n
)
+
∑
k
=
0
n
−
1
z
k
(
k
+
a
)
s
and
Φ
(
z
,
s
−
1
,
a
)
=
(
a
+
z
∂
∂
z
)
Φ
(
z
,
s
,
a
)
and
Φ
(
z
,
s
+
1
,
a
)
=
−
1
s
∂
∂
a
Φ
(
z
,
s
,
a
)
.
A series representation for the Lerch transcendent is given by
Φ
(
z
,
s
,
q
)
=
1
1
−
z
∑
n
=
0
∞
(
−
z
1
−
z
)
n
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
q
+
k
)
−
s
.
(Note that
(
n
k
)
is a binomial coefficient.)
The series is valid for all s, and for complex z with Re(z)<1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function.
A Taylor's series in the first parameter was given by Erdélyi. It may be written as the following series, which is valid for
|
log
(
z
)
|
<
2
π
;
s
≠
1
,
2
,
3
,
…
;
a
≠
0
,
−
1
,
−
2
,
…
Φ
(
z
,
s
,
a
)
=
z
−
a
[
Γ
(
1
−
s
)
(
−
log
(
z
)
)
s
−
1
+
∑
k
=
0
∞
ζ
(
s
−
k
,
a
)
log
k
(
z
)
k
!
]
B. R. Johnson (1974). "Generalized Lerch zeta-function". Pacific J. Math. 53 (1): 189–193.
If s is a positive integer, then
Φ
(
z
,
n
,
a
)
=
z
−
a
{
∑
k
=
0
k
≠
n
−
1
∞
ζ
(
n
−
k
,
a
)
log
k
(
z
)
k
!
+
[
ψ
(
n
)
−
ψ
(
a
)
−
log
(
−
log
(
z
)
)
]
log
n
−
1
(
z
)
(
n
−
1
)
!
}
,
where
ψ
(
n
)
is the digamma function.
A Taylor series in the third variable is given by
Φ
(
z
,
s
,
a
+
x
)
=
∑
k
=
0
∞
Φ
(
z
,
s
+
k
,
a
)
(
s
)
k
(
−
x
)
k
k
!
;
|
x
|
<
ℜ
(
a
)
,
where
(
s
)
k
is the Pochhammer symbol.
Series at a = -n is given by
Φ
(
z
,
s
,
a
)
=
∑
k
=
0
n
z
k
(
a
+
k
)
s
+
z
n
∑
m
=
0
∞
(
1
−
m
−
s
)
m
Li
s
+
m
(
z
)
(
a
+
n
)
m
m
!
;
a
→
−
n
A special case for n = 0 has the following series
Φ
(
z
,
s
,
a
)
=
1
a
s
+
∑
m
=
0
∞
(
1
−
m
−
s
)
m
Li
s
+
m
(
z
)
a
m
m
!
;
|
a
|
<
1
,
where
Li
s
(
z
)
is the polylogarithm.
An asymptotic series for
s
→
−
∞
Φ
(
z
,
s
,
a
)
=
z
−
a
Γ
(
1
−
s
)
∑
k
=
−
∞
∞
[
2
k
π
i
−
log
(
z
)
]
s
−
1
e
2
k
π
a
i
for
|
a
|
<
1
;
ℜ
(
s
)
<
0
;
z
∉
(
−
∞
,
0
)
and
Φ
(
−
z
,
s
,
a
)
=
z
−
a
Γ
(
1
−
s
)
∑
k
=
−
∞
∞
[
(
2
k
+
1
)
π
i
−
log
(
z
)
]
s
−
1
e
(
2
k
+
1
)
π
a
i
for
|
a
|
<
1
;
ℜ
(
s
)
<
0
;
z
∉
(
0
,
∞
)
.
An asymptotic series in the incomplete Gamma function
Φ
(
z
,
s
,
a
)
=
1
2
a
s
+
1
z
a
∑
k
=
1
∞
e
−
2
π
i
(
k
−
1
)
a
Γ
(
1
−
s
,
a
(
−
2
π
i
(
k
−
1
)
−
log
(
z
)
)
)
(
−
2
π
i
(
k
−
1
)
−
log
(
z
)
)
1
−
s
+
e
2
π
i
k
a
Γ
(
1
−
s
,
a
(
2
π
i
k
−
log
(
z
)
)
)
(
2
π
i
k
−
log
(
z
)
)
1
−
s
for
|
a
|
<
1
;
ℜ
(
s
)
<
0.
The Lerch transcendent is implemented as LerchPhi in Maple.
