HAL: hal-00693281, version 1

arXiv: 1205.0636

Detailed view

Export this paper BibTeX - EndNote - TEI - RefWorks -

Available versions:

v1 (2012-05-03)

v2 (2013-03-18)

A Necessary Condition for a Nontrivial Zero of the Riemann Zeta Function via the Polylogarithmic Function

Lazhar Fekih-Ahmed 1

(2012-05-02)

We provide a new series expansion of the polylogarithm of complex argument $\Li_{s}(x)=\sum_{n=1}^{\infty}\frac{x^n}{n^{s}}$. From the new series, we define a new entire function $Z(s,x)$ which is related to $\Li_{s}(x)$ but processes several advantages over the initial polylogarithmic series. For example, the limit of $Z(s,x)$ when $x\to 1$ is convergent to $(s-1)\zeta(s)$ for all complex numbers $s$ while le limit of $\Li_{s}(x)$ converges only when $\re(s)>1$. As an application of the expansion of $Z(s,x)$, we derive of a necessary condition for a non-trivial zero of the Riemann zeta function.

1:	Ecole Nationale d'Ingénieurs de Tunis (ENIT)
	Ecole Nationale d'Ingénieurs de Tunis

Subject

:

Mathematics/Number Theory

Keyword(s): Number Theory – Polylogarithm function – Riemann Zeta function – Riemann nontrivial zeros

hal-00693281, version 1

http://hal.archives-ouvertes.fr/hal-00693281

oai:hal.archives-ouvertes.fr:hal-00693281

From: Lazhar Fekih-Ahmed <>

Submitted on: Wednesday, 2 May 2012 12:14:52

Updated on: Thursday, 3 May 2012 09:04:04