An entire function defined by Riemann

TL;DR

This paper explores a unique entire function linked to Riemann's zeta function, providing new integral representations and proofs, and establishing connections with the zeros of the zeta function.

Contribution

It introduces a novel integral representation of the zeta function and offers proofs of Riemann's formulas, expanding understanding of the zeta function's properties.

Findings

Established an integral representation of the zeta function different from known forms.

Connected an entire function to the zeros of the zeta function.

Provided proofs of Riemann's formulas related to the zeta function.

Abstract

In one of the sheets in Riemann's Nachlass he defines an entire function and connect it with his zeta function. As in many pages in his Nachlass, Riemann is not giving complete proofs. However, I consider that this work is undoubtedly by Riemann. He obtains an $L^\infty$ function whose Fourier transform vanish at the real values $\gamma$ with $\zeta(\frac12+i\gamma)=0$. We give proofs of Riemann formulas. This is an integral representation of the zeta function different from the known ones. I believe this is the first time it has been published.