Andrew Granville's Home Page

2018 Publications

Large Character Sums: Burgess's theorem and zeros of L-functions
(with K. Soundararajan)
Journal of the European Mathematical Society, 20 (2018), 1-14

We study the conjecture that \( \sum_{n\leq x} \chi(n)=o(x)\) for any primitive Dirichlet character \( \chi \pmod q\) with \( x\geq q^\epsilon\), which is known to be true if the Riemann Hypothesis holds for \( L(s,\chi)\). We show that it holds under the weaker assumption that `\(100\%\)' of the zeros of \( L(s,\chi)\) up to height \( \tfrac 14\) lie on the critical line. We also establish various other consequences of having large character sums; for example, that if the conjecture holds for \( \chi^2\) then it also holds for \( \chi\).

Article and Journal

The frequency and the structure of large character sums
(with Jonathan Bober, Leo Goldmakher and Dimitris Koukoulopoulos)
Journal of the European Mathematical Society, 20 (2018), 1759-1818

Let \( M(\chi)\) denote the maximum of \( |\sum_{n\le N}\chi(n)|\) for a given non-principal Dirichlet character \( \chi \pmod q\), and let \( N_\chi\) denote a point at which the maximum is attained. In this article we study the distribution of \( M(\chi)/\sqrt{q}\) as one varies over characters \( \pmod q\), where \( q\) is prime, and investigate the location of \( N_\chi\). We show that the distribution of \( M(\chi)/\sqrt{q}\) converges weakly to a universal distribution \( \Phi\), uniformly throughout most of the possible range, and get (doubly exponential decay) estimates for \( \Phi\)'s tail. Almost all \( \chi\) for which \( M(\chi)\) is large are odd characters that are \( 1\)-pretentious. Now, \( M(\chi)\ge |\sum_{n\le q/2}\chi(n)| = \frac{|2-\chi(2)|}\pi \sqrt{q} |L(1,\chi)|\), and one knows how often the latter expression is large, which has been how earlier lower bounds on \( \Phi\) were mostly proved. We show, though, that for most \( \chi\) with \( M(\chi)\) large, \( N_\chi\) is bounded away from \( q/2\), and the value of \( M(\chi)\) is little bit larger than \( \frac{\sqrt{q}}{\pi} |L(1,\chi)|\).

Article and Journal

Using Dynamical Systems to Construct Infinitely Many Primes,
American Mathematical Monthly 125 (2018), 483-496

Euclid's proof can be reworked to construct infinitely many primes, in many different ways, using ideas from arithmetic dynamics.

Article and Journal

A more intuitive proof of a sharp version of Halasz's theorem
(with Adam Harper and K. Soundararajan)
Proceedings of the American Mathematical Society 146 (2018), 4099-4104

We prove a sharp version of Halasz's theorem on sums \( \sum_{n \leq x} f(n)\) of multiplicative functions \( f\) with \( |f(n)|\le 1\). Our proof avoids the ``average of averages'' and ``integration over \( \alpha\)'' manoeuvres that are present in many of the existing arguments. Instead, motivated by the circle method we express \( \sum_{n \leq x} f(n)\) as a triple Dirichlet convolution, and apply Perron's formula.

Article and Journal

When does the Bombieri-Vinogradov Theorem hold for a given multiplicative function?
(with Xuancheng (Fernando) Shao)
Forum of Mathematics, Sigma 6 (2018), e15, 23 pages.

Let \( f\) and \( g\) be \( 1\)-bounded multiplicative functions for which \( f*g=1_{.=1}\). The Bombieri-Vinogradov Theorem holds for both \( f\) and \( g\) if and only if the Siegel-Walfisz criterion holds for both \( f\) and \( g\), and the Bombieri-Vinogradov Theorem holds for \( f\) restricted to the primes.