§25.11 Hurwitz Zeta Function
Contents
- §25.11(i) Definition
- §25.11(ii) Graphics
- §25.11(iii) Representations by the Euler–Maclaurin Formula
- §25.11(iv) Series Representations
- §25.11(v) Special Values
- §25.11(vi) Derivatives
- §25.11(vii) Integral Representations
- §25.11(viii) Further Integral Representations
- §25.11(ix) Integrals
- §25.11(x) Further Series Representations
- §25.11(xi) Sums
- §25.11(xii)
-Asymptotic Behavior
§25.11(i) Definition
The function
was introduced in Hurwitz (1882)
and defined by the series expansion
has a meromorphic continuation in the
-plane, its only
singularity in
being a simple pole at
with residue 1.
As a function of
, with
(
) fixed,
is analytic in the half-plane
. The
Riemann zeta function is a special case:
For most purposes it suffices to restrict
because of
the following straightforward consequences of (25.11.1):
Most references treat real
with
.
§25.11(iii) Representations by the Euler–Maclaurin Formula
§25.11(iv) Series Representations
§25.11(v) Special Values
Throughout this subsection
.
§25.11(vi) Derivatives
¶
-Derivative
¶
-Derivatives
In (25.11.18)–(25.11.24) primes on
denote derivatives with respect to
. Similarly in §§25.11(viii)
and 25.11(xii).
where
are integers with
and
.
§25.11(vii) Integral Representations
where the integration contour is a loop around the negative real axis as described for (25.5.20).
§25.11(viii) Further Integral Representations
where
§25.11(ix) Integrals
§25.11(xi) Sums
where
is Catalan’s constant:
§25.11(xii)
-Asymptotic Behavior
As
with
fixed,
As
with
fixed,
,
uniformly with respect to bounded nonnegative values of
.
As
in the sector
, with
and
fixed, we have the asymptotic expansion
Similarly, as
in the sector
,
and
For the more general case
,
, see Elizalde (1986).












