Formula:DLMF:25.13:E1

$\displaystyle {\displaystyle \PeriodicZeta@{x}{s} = \sum_{n=1}^\infty \frac{\expe^{2 \cpi \iunit n x}}{n^s} }$

Constraint(s)

$\displaystyle {\displaystyle x}$ real &
$\displaystyle {\displaystyle \realpart{s} > 1}$ if $\displaystyle {\displaystyle x}$ is an integer, $\displaystyle {\displaystyle \realpart{s} > 0}$ otherwise

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

& : logical and
$\displaystyle {\displaystyle F}$  : periodic zeta function : http://dlmf.nist.gov/25.13#E1
$\displaystyle {\displaystyle \Sigma}$  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
$\displaystyle {\displaystyle \mathrm{e}}$  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
$\displaystyle {\displaystyle \pi}$  : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
$\displaystyle {\displaystyle \mathrm{i}}$  : imaginary unit : http://dlmf.nist.gov/1.9.i
$\displaystyle {\displaystyle \Re {z}}$  : real part : http://dlmf.nist.gov/1.9#E2