Exponential sums of Lerch’s zeta functions

Abstract:

For $x$ not an an integer and $\operatorname {Re} (s) > 0$, let \[ F(x,s) = \sum \limits _{k = 1}^\infty {\frac {{{e^{2\pi ikx}}}}{{{k^s}}}} \] be the Lerch’s zeta function. In this note, we will show that \[ \sum \limits _{\gamma = 1}^{m - 1} {{e^{ - 2\pi i\gamma \alpha /m}}} F\left ( {\frac {\gamma }{m},1 - n} \right ) = \frac {1}{n}\left ( {{B_n} - {m^n}{B_n}\left ( {\frac {\alpha }{m} - \left [ {\frac {\alpha }{m}} \right ]} \right )} \right )\] where $\alpha$ is an integer and $\alpha \not \equiv 0(\mod m)$ and $n \geqslant 1$. For $n = 1$, this formula is equivalent to the classical Eisenstein formula \[ \frac {\alpha }{m} - \left [ {\frac {\alpha }{m}} \right ] - \frac {1}{2} = - \frac {1}{{2m}}\sum \limits _{\gamma = 1}^{m - 1} {\sin \frac {{2\pi \gamma \alpha }}{m}} \cot \frac {{\pi \gamma }}{n}.\]