Lerch zeta function

Summary

The Lerch zeta function is the series $\sum_{n \ge 0} e^{2\pi i \lambda n} (n + \alpha)^{-s}$, a generalized Dirichlet series with a phase factor and a shift. In the closely related form $\Phi(z, s, a) = \sum_{n \ge 0} z^n (n + a)^{-s}$ it is called the Lerch transcendent. It contains the Hurwitz zeta function, the Riemann zeta function, and the polylogarithm as special cases. Dirichlet $L$-functions are finite linear combinations of its values. So identities proved for it hold for all of them at once.