The Riemann hypothesis in characteristic \(p\) in historical perspective
type: Item
instance of: scholarly article
title: The Riemann hypothesis in characteristic \(p\) in historical perspective
zbMATH Open document ID: 1414.11003
DOI: 10.1007/978-3-319-99067-5
author: Peter Roquette
published in: Lecture Notes in Mathematics
publication date: 2018-08-28T00:00:00Z
review text: The book deals with the history of the zeros of the zeta function of a global function field. Let \(k={\mathbb F}_q\) be the finite field of \(q=p^r\) elements and let \(K/k\) be a function field with exact field of constants \(k\). The zeta function of \(K\) is the function \[\zeta_K(s)= \prod_P \frac{1}{1-|P|^{-s}}=\sum_A |A|^{-s}\] where \(P\) ranges over the primes of \(K\) and \(A\) over the integer divisors of \(K\). Here \(|P|\) denotes the number of elements in the residue field corresponding to \(P\) and \(s\in {\mathbb C}\) satisfies \(\mathrm{Re}( s)>1\). The function \(\zeta_K(s)\) can be extended to a meromorphic function of \({\mathbb C}\) with poles of order \(1\) at \(s=0\) and \(s=1\). The Riemann hypothesis (RH) says that all zeros of \(\zeta_K(s)\) belong to the line with real part \(\mathrm{Re}( s)=\frac{1}{2}\). The RH was considered by Riemann for the field of rational numbers \({\mathbb Q}\) and the generalization of the zeta function to a general number field is known as the Dedekind zeta function of the field. The RH is an open problem in characteristic \(0\). In characteristic \(p>0\) the RH was finally settled by \textit{André Weil} in 1940--1941 [C. R. Acad. Sci., Paris 210, 592--594 (1940; Zbl 0023.29401); Proc. Natl. Acad. Sci. USA 27, 345--347 (1941; Zbl 0061.06406)]. Roquette's book is not just the history of Weil's proof of the RH. It is much more. It is a detailed and accurate history of the RH in characteristic \(p\) and it is also a description on how and when the main protagonists at the time made their corresponding contributions towards the proof of the RH. It is worth to point out that the author met most of the protagonists of the solution of the problem. The book is very pleasant to read and should be consulted by any one interested in history, in function fields or in general in the RH in any characteristic. The book can be used by specialists and by non-specialists as a brief but very interesting introduction to function f…
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11-03
Mathematics Subject Classification ID: 14-03
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 01A80
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 6928403
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: global function fields
zbMATH Keywords: zeta function in characteristic \(p\), Emil Artin
zbMATH Keywords: Helmut Hasse
zbMATH Keywords: André Weil
zbMATH Keywords: Friedrich Karl Schmidt
zbMATH Keywords: Max Deuring
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-99067-5
OpenAlex ID: W2894099048
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description: scientific article; zbMATH DE number 6928403