# The Riemann Hypothesis: A Qualitative Characterization of the Nontrivial Zeros of the Riemann Zeta Function Using Polylogarithms

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Abstract : We formulate a parametrized uniformly absolutely globally convergent series of ζ(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s − 1)ζ(s) + 1 x Li s z z − 1 dz, where Li s (x) is the polylogarithm function. As an immediate first application of the new parametrized series, a new expression of ζ(s) follows: (s − 1)ζ(s) = − 1 0 Li s z z − 1 dz. As a second important application, using the functional equation and exploiting uniform convergence of the series defining Z(s, x), we have for any non-trivial zero s
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https://hal.archives-ouvertes.fr/hal-01355277
Contributor : Lazhar Fekih-Ahmed <>
Submitted on : Tuesday, August 23, 2016 - 3:41:20 PM
Last modification on : Thursday, August 25, 2016 - 1:01:06 AM
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• HAL Id : hal-01355277, version 1
• ARXIV : 1608.06737

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Lazhar Fekih-Ahmed. The Riemann Hypothesis: A Qualitative Characterization of the Nontrivial Zeros of the Riemann Zeta Function Using Polylogarithms. 2016. 〈hal-01355277〉

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