### 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.

Article