[Paper Review] On Riemanns Nachlass for Analytic Number Theory: A translation of Siegel's Uber

Eric Barkan, David P. Sklar|arXiv (Cornell University)|Oct 11, 2018

History and Theory of Mathematics4 citations

TL;DR

This paper presents a translation and analysis of Carl Ludwig Siegel's 1932 work on Riemann's unpublished notes in analytic number theory, revealing two major results: the Riemann-Siegel formula for asymptotic expansion of the Riemann zeta function on the critical line, and a new integral representation of the zeta function. The key contribution is the recovery and formalization of Riemann’s deep analytic techniques, particularly through the use of a special theta-type integral, which enabled efficient computation and new zero-density estimates.

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 In the early 1930's two related unpublished results from 1859 were found in Riemanns very rough notes by the great twentieth century mathematician and scholar, Carl Ludwig Siegel (1896-1981).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Nachlass zur analytischen Zahlentheorie"("On Riemann's Nachlass for Analytic Number Theory") Siegel presents these unpublished results and gives derivations he found in Riemann's notes.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.

Motivation & Objective

• To recover and interpret Riemann’s unpublished notes on the zeta function, particularly those related to analytic number theory.
• To clarify the origin and derivation of the Riemann-Siegel formula and the integral representation of the zeta function as found in Riemann’s fragments.
• To demonstrate that Riemann’s unpublished work contained advanced analytic techniques, including asymptotic expansions and integral transforms, that predated and anticipated later results by Hardy and Littlewood.
• To provide a rigorous, translated exposition of Siegel’s 1932 paper, making it accessible to modern researchers and preserving the mathematical integrity of Riemann’s original insights.

Proposed method

• Derivation of the Riemann-Siegel formula using the saddle-point method applied to an integral transform involving the function $\varPhi(\tau, u)$.
• Use of Cauchy’s theorem to derive functional equations for $\varPhi(u)$, which underlie the asymptotic expansion of $\zeta(s)$.
• Application of the integral representation of $\zeta(s)$ in terms of $\varPhi(\tau, u)$ to derive asymptotic behavior on the critical line $\sigma = \frac{1}{2}$.
• Estimation of the number of zeros of $\zeta(s)$ in the critical strip using bounds derived from the argument principle and the asymptotic expansion.
• Use of the functional equation and contour integration techniques to relate the zeta function to the theta-like integral $\varPhi(\tau, u)$.
• Establishment of a lower bound for the number of zeros $N_0(T)$ on the critical line by analyzing the growth of $N_2(T)$, the number of zeros in a shifted strip.

Experimental results

Research questions

• RQ1 How did Riemann derive the asymptotic expansion of the zeta function on the critical line, and what analytic tools did he use?
• RQ2 What is the mathematical significance of the integral $\varPhi(\tau, u)$ in the context of the zeta function and its zeros?
• RQ3 To what extent did Riemann’s unpublished notes contain results that predated or anticipated later work by Hardy and Littlewood?
• RQ4 Can the Riemann-Siegel formula be rigorously derived from Riemann’s notes, and what gaps, if any, did Siegel need to fill?
• RQ5 What can be inferred about the distribution of non-trivial zeros of the zeta function from Riemann’s fragmentary derivations?

Key findings

• The Riemann-Siegel formula provides an efficient asymptotic expansion for $\zeta(s)$ on the critical line, with the principal term rediscovered by Hardy and Littlewood in 1920.
• The integral $\varPhi(\tau, u)$, central to the derivation, satisfies functional equations that allow systematic generation of higher-order terms in the asymptotic series.
• A lower bound for the number of non-trivial zeros $N_0(T)$ on the critical line is established as $N_0(T) > \frac{3}{8\pi}e^{-3/2}T + \frac{3}{2}e^{-3/2}\sum_{\alpha_\zeta > 1/2}(\alpha_\zeta - \frac{1}{2}) + o(T)$, implying a positive density greater than $1/38$.
• The method yields a new zero-density result: in the region $0 \leq \sigma - \frac{1}{2} \leq \frac{\psi(t)}{\log t}$ for $2 \leq t \leq T$, there are at least $\frac{3}{4\pi}e^{-3/2}T\psi(T)(1+o(1))$ zeros, even when $\psi(t)$ grows slower than $\log \log t$.
• The sum $\sum (\alpha_\zeta - \frac{1}{2})$ over zeros to the right of the critical line is bounded, and if it grows faster than $T$, the Riemann Hypothesis cannot be 'too false'.
• The paper confirms that Riemann’s unpublished work contained sophisticated analytic techniques, particularly in asymptotic analysis and contour integration, that were not fully appreciated until Siegel’s rediscovery in 1932.