Report on some papers related to the function $\mathop{\mathcal R }(s)$ found by Siegel in Riemann's posthumous papers

TL;DR

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Contribution

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Abstract

In a letter to Weierstrass Riemann asserted that the number $N_0(T)$ of zeros of $\zeta(s)$ on the critical line to height $T$ is approximately equal to the total number of zeros to this height $N(T)$. Siegel studied some posthumous papers of Riemann trying to find a proof of this. He found a function $\mathop{\mathcal R }(s)$ whose zeros are related to the zeros of the function $\zeta(s)$. Siegel concluded that Riemann's papers contained no ideas for a proof of his assertion, connected the position of the zeros of $\mathop{\mathcal R }(s)$ with the position of the zeros of $\zeta(s)$ and asked about the position of the zeros of $\mathop{\mathcal R }(s)$. This paper is a summary of several papers that we will soon upload to arXiv, in which we try to answer Siegel's question about the position of the zeros of $\mathop{\mathcal R }(s)$. The articles contain also improvements on Siegel's results and also other possible ways to prove Riemann's assertion, but without achieving this goal.