An Analytical and Numerical Detour for the Riemann Hypothesis

Abstract

From the functional equation F(s)=F(1−s) of Riemann’s zeta function, this article gives new insight into Hadamard’s product formula. The function S1(s)=d(lnF(s))/ds and its family of associated Sm functions, expressed as a sum of rational fractions, are interpreted as meromorphic functions whose poles are the poles and zeros of the F function. This family is a mathematical and numerical tool which makes it possible to estimate the value F(s) of the function at a point s=x+iy=x˙+½+iy in the critical strip S from a point 𝓈=½+iy on the critical line ℒ.Generating estimates Sm∗(s) of Sm(s) at a given point requires a large number of adjacent zeros, due to the slow convergence of the series. The process allows a numerical approach of the Riemann hypothesis (RH). The method can be extended to other meromorphic functions, in the neighborhood of isolated zeros, inspired by the Weierstraß canonical form. A final and brief comparison is made with the ζ and F functions over finite fields.

Keywords

• Riemann hypothesis (RH)
• functional equation
• meromorphic function
• Weierstraß factorization