A 2-variable power series approach to the Riemann hypothesis

TL;DR

This paper introduces a novel two-variable power series approach to the Riemann hypothesis, linking the zeros of a specific series to the hypothesis and establishing equivalences involving a constant T similar to de Bruijn-Newman.

Contribution

It develops a new two-variable power series framework for analyzing the Riemann hypothesis and proves key equivalences involving the zeros and a constant T.

Findings

Zeros of B_y(f^b) are inverses of a real-coefficient power series.

Existence of a constant T with T=1 equivalent to the Riemann hypothesis.

Riemann hypothesis implies zeros of B_y(f^b) are simple for 0<y<1.

Abstract

We consider the power series in two complex variables By(fb)(x)=S_(n=0)|.A_n^b x^n y^(n(n+1)/2)., where .(-1).^n A_n^b are the non-zero coefficients of the Maclaurin series of the Riemann Xi function. The Riemann hypothesis is the assertion that all zeros of B_1 (f^b) are real. We prove that every zero of B_y (f^b) is the inverse of a power series in y with real coefficients, which converges for |y|<0,2078.... We show the existence of a constant T, similar to the de Bruijn-Newman constant, satisfying : 0= y =T if and only if all zeros of B_y (f^b) are real. We prove that 1/4 = T = 1 and that T=1 is equivalent to the Riemann hypothesis. We show that the Riemann hypothesis is also equivalent to what the discriminant of each Jensen polynomial of B_y (f^b) does not vanish on the interval [1/4,1|[. We prove the Riemann hypothesis implies that the zeros of B_y (f^b) are simple for 0<y<1, and we conjecture that the reciprocal implication is true.