  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 In the early 1930's two related unpublished results from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unpublished results and gives derivations he found in Riemann's notes.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.
