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

Lazhar Fekih-Ahmed (1)

1 ENIT - Ecole Nationale d'Ingénieurs de Tunis

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

Mots clés: Polylogarithms; Riemann Hypothesis; Number Theory; Riemann Zeta function

Domaines: Théorie des nombres [math.NT]

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