On Riemanns Nachlass for Analytic Number Theory: A translation of Siegel's Uber

Abstract

In 1859 Riemann (1826-1866) published his only paper on number theory. In this eight-page paper he obtained a formula for the number of primes less than or equal to a real number x, and revealed the deep connection between the distribution of primes and the zeros of an analytic function now called the Riemann Zeta Function. In the early 1930's two related unpublished results from 1859 were found in Riemanns very rough notes by the great twentieth century mathematician and scholar, Carl Ludwig Siegel (1896-1981). In his 1932 paper "Uber Riemanns Nachlass zur analytischen Zahlentheorie"("On Riemann's Nachlass for Analytic Number Theory") Siegel presents these unpublished results and gives derivations he found in Riemann's notes. The first is an asymptotic development, now called the Riemann-Siegel formula, for efficiently computing values of the Riemann Zeta Function. The second is a new integral representation of the zeta function. These results had not been rediscovered seventy years after Riemann. Thus, in 1932 the importance of Siegel's paper was not only its contribution to the history of mathematics, but also its contribution to current research. Hoping to get some insight into how Siegel spotted and deciphered these gems among Riemann's fragmentary and disordered personnel papers, we decided to look at the 1932 paper. We also wanted to learn how much of the paper is original to Riemann and whether Siegel needed to fill any gaps. Although Siegel's paper is cited whenever the Riemann-Siegel formula is discussed, we were unable the locate an English translation. Despite our limited knowledge of the German language, we have produced a translation of the paper as it appears in volume one of Siegel's collected works.