Andrew Granville's Home Page

Preprints accepted for publication

The frequency and the structure of large character sums
(with Jonathan Bober, Leo Goldmakher and Dimitris Koukoulopoulos)
to appear in the Journal of the European Mathematical Society

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

Large Character Sums: Burgess's theorem and zeros of L-functions
(with K. Soundararajan)
to appear in the Journal of the European Mathematical Society

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

Using Dynamical Systems to Construct Infinitely Many Primes
to appear in the American Mathematical Monthly

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

Article

A more intuitive proof of a sharp version of Halasz's theorem
(with Adam Harper and K. Soundararajan)
to appear in the Proceedings of the American Mathematical Society

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

Beyond the LSD method for the partial sums of multiplicative functions
(with Dimitris Koukoulopoulos)
to appear in the Ramanujan Journal

The Landau-Selberg-Delange (LSD) method gives an asymptotic formula for the partial sums of a multiplicative function \( f\) whose prime values are \(\alpha\) on average. In the literature, the average is usually taken to be \(\alpha\) with a very strong error term, leading to an asymptotic formula for the partial sums with a very strong error term. In practice, the average at the prime values may only be known with a fairly weak error term, and so we explore here how good an estimate this will imply for the partial sums of \(f\) , developing new techniques to do so.

Article