Pretentiousness in analytic number theory
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.
mean values of totally multiplicative functions
multiplicative functions in arithmetic progressions
scientific article; zbMATH DE number 5620675
scientific article