Hurwitz Zeta Function

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Hurwitz Zeta Function

Definition

Constraint(s): &




Constraint(s):


Representations by the Euler-Maclaurin Formula

Constraint(s): &
&
&


Constraint(s): &
&


Constraint(s): &
&
&


Series Representations

Constraint(s): &
&


Constraint(s): &


Constraint(s): &


Special Values

Constraint(s):


Constraint(s): &


Constraint(s):


Constraint(s): &


Constraint(s): &
&


Constraint(s): &
integers, &


Derivatives

a-Derivative

Constraint(s): &


s-Derivatives

Constraint(s):


Constraint(s): &
&


Constraint(s): &
&


Constraint(s): are integers with &


Constraint(s):


Constraint(s):


Constraint(s): &


Integral Representations

Constraint(s): &


Constraint(s): &


Constraint(s): &
&


Constraint(s): &
&


Constraint(s): &


Constraint(s): &
&

the integration contour is a loop around the negative real axis; it starts at , encircles the origin once in the positive direction without enclosing any of the points

, and returns to


Further Integral Representations

Constraint(s): &


Substitution(s):


Constraint(s): &


Constraint(s): &


Further Series Representations

Constraint(s): , &
or , ,


Constraint(s): &
is a Dirichlet character~


Sums

Constraint(s): &


Constraint(s): &
&


a-Asymptotic Behavior

Constraint(s): with fixed


Constraint(s): with fixed &


Constraint(s): in the sector , with and fixed


Constraint(s): in the sector