Introduction
In this brief Insight article the analytic continuations of the Lerch Transcendent and Riemann Zeta Functions are achieved via the Euler’s Series Transformation (zeta) and a generalization thereof, the E-process (Lerch). Dirichlet Series is mentioned as a steppingstone. The continuations are given but not shown to be convergent by any means, though if you the reader would be interested in such write me in the comments and I may oblige with an update if I get around to it. Some basic complex analysis and (double) series manipulations are the only assumed knowledge herein.
Euler’s Series Transformation and the E-Process
We wish to consider the supposed convergent alternating series ##\sum\limits_{k = 1}^\infty {{{\left( { – 1} \right)}^{k – 1}}{a_k}} ## by use of the power series
$$f\left( x \right) = \sum\limits_{k = 1}^\infty {{{\left( { – 1} \right)}^{k – 1}}{a_k}} {x^k}\text{ (1.1) }$$
Which we require to converge for at least ## – 1 < x \leq 1## . That we...