Riemann Zeta Function Expressed as the Difference of Two Symmetrized Factorials Whose Zeros All Have Real Part of 1/2

TL;DR

This paper presents new series and polynomial representations of the Riemann zeta function, demonstrating that it can be expressed as the difference of two symmetrized factorials with zeros on the critical line, supporting the Riemann hypothesis.

Contribution

It introduces a novel expression of ta as a difference of symmetrized factorials with zeros on the critical line, providing new insights into the Riemann hypothesis.

Findings

ta expressed as a difference of two symmetrized factorials with zeros on Re(s)=1/2

Interlacing zeros of the factorials support the Riemann hypothesis

New series representations enable fast ta computation

Abstract

In this paper, some new results are reported for the study of Riemann zeta function $\zeta(s)$ in the critical strip $0<Re(s)<1$, such as $\zeta(s)$ expressed in a generalized Euler product only involving prime numbers. Particularly, some new absolutely convergent series representations of $\zeta(s)$ based on binomial expansion are presented. The crucial progress is to find that $\zeta(s)$ can be expressed as a linear combination of polynomials of infinite degree, whose consequences are shown in several aspects: (i) numerically it provides a scenario to construct very fast convergent algorithm to calculate $\zeta(s)$; (ii) interestingly it shows that Lagrange interpolation using infinite number of integer Euler zeta functions reproduces the exact complex $\zeta(s)$; (iii) surprisingly it demonstrates that alternating Riemann zeta function (or other entire functions removing the pole of zeta function) is admissible to Melzak combinatorial transform for polynomials. Applying the functional symmetry on $\zeta(s)$ in the form of Melzak transform induces $\zeta(s)$ being written as the difference of two symmetrized factorials whose zeros are proved to all have real part of 1/2. Furthermore, the two symmetrized factorials are proved to have interlacing between the two sequences of the imaginary part of their zeros on upper (or lower) half plane, which ensures the difference of the two symmetrized factorials [proportional to $\zeta(s)$] attaining the same feature of zeros with real part of 1/2 to endorse Riemann hypothesis.