An entire function defined by Riemann

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.