The Analytic Continuation of the Lerch and the Zeta Functions

[benorin]

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...

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[jvicens]

Hello,

Thank you for sharing your insights on the analytic continuations of the Lerch Transcendent and Riemann Zeta Functions through the use of Euler's Series Transformation and the E-Process. It is fascinating to see how these techniques can be used to extend the domain of convergence for these important functions.

I am always interested in learning more about complex analysis and series manipulations. It would be great if you could provide some more details or examples that demonstrate the convergence of these series. This would help readers like myself to better understand the validity of your results.

Thank you again for sharing your research and I look forward to any updates or additional information you may provide in the future.