Pretentiousness in analytic number theory
type: Item
instance of: scholarly article
title: Pretentiousness in analytic number theory
zbMATH Open document ID: 1236.11086
author: Andrew Granville
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-10-26T00:00:00Z
full work available at URL: https://eudml.org/doc/10868
review text: This interesting article is a wide ranging survey of methods and results, both classical and recent, in analytic number theory. Let \(f\), \(g\) be two totally multiplicative functions from \(\mathbb N\) to the complex unit disc. Then \(f(n)\) is said to ``pretend'' to be \(g(n)\) if the sum \[ \sum_{p\leq x}{1\over p} (1- \text{Re}(f(p)\,\overline g(p)) \] is bounded. The author describes how the idea of pretentiousness can play a useful role in the study of various well-known problems. In this context he discusses in particular the prime number theorem, large character sums, sums over arithmetic progressions and exponential sums. He quotes several results from the paper [``Multiplicative Functions in Arithmetic Progressions'', to appear, \url{http://arxiv.org/abs/math/0702389}] by \textit{A. Balog, A. Granville} and \textit{K. Soundararajan}. One of these theorems states that if the mean value of \(f\) is ``large'' in an arithmetic progression mod \(q\), then \(f(n)\) pretends to be \(\chi(n)n^{it}\) for some Dirichlet character \(\chi\bmod q\) and some ``small'' real \(t\). Several related papers by the author and \textit{K. Soundararajan} are listed in the references and referred to in the text. In particular in [J. Am. Math. Soc. 20, 357--384 (2007; Zbl 1210.11090] they prove that if the character sum \(|\sum_{n\leq x}\chi(n)|\) is ``large'', where \(\chi\) is a character mod \(q\), then \(\chi\) pretends to be a character \(\psi\bmod m\) for \(m\leq (\log q)^{1/3}\). Various other key results (too complicated to state here) are described and discussed.
reviewed by: Eira J. Scourfield
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11N13
zbMATH DE Number: 5620675
zbMATH Keywords: pretentiousness
zbMATH Keywords: prime number theorem
zbMATH Keywords: character sums
zbMATH Keywords: mean values of totally multiplicative functions
zbMATH Keywords: multiplicative functions in arithmetic progressions
MaRDI profile type: Publication
OpenAlex ID: W2325605712
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DOI: 10.5802/JTNB.664
description: scientific article; zbMATH DE number 5620675