Both real and imaginary parts of the function F(s)=zeta(s)*Gamma(s/2)*pi**(-s/2)=xi(s)/(s*(s-1)), whose zeroes exactly coincide with the non-trivial zeroes of the Riemann zeta-function, have infinitely many zeroes for any value of Re(s)

TL;DR

This paper proves that both the real and imaginary parts of a function related to the Riemann zeta-function have infinitely many zeroes for any real part of s, extending understanding of zero distribution.

Contribution

It establishes that both parts of the function F(s), linked to the Riemann zeta-function, have infinitely many zeroes for all Re(s), regardless of the value.

Findings

Both parts of F(s) have infinitely many zeroes for any Re(s)

Zeroes of F(s) coincide with non-trivial zeroes of zeta(s)

Supports the universality of zero distribution in complex analysis

Abstract

The following theorem is proven: Both real and imaginary parts of the function F(s) defined as F(s)=zeta(s)*Gamma(s/2)*pi**(-s/2)=xi(s)/(s*(s-1)), and whose zeroes exactly coincide with the non-trivial zeroes of the Riemann zeta-function, have infinitely many zeroes for any value of Re(s).