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In this Monthly note, we use Van der Waerden's Theorem and Fermat's Theorem on four squares in an arithmetic progression to prove that there are infinitely many primes.

(with Jonathan Bober, Leo Goldmakher and Dimitris Koukoulopoulos)

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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 \( \frac{\sqrt{q}}{\pi} |L(1,\chi)|\).

(with Sary Drappeau and Xuancheng (Fernando) Shao)

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We show that smooth-supported multiplicative functions \( f\) are well-distributed in arithmetic progressions \( a_1a_2^{-1} \pmod q\) on average over moduli \( q\leq x^{3/5-\varepsilon}\) with \( (q,a_1a_2)=1\) . This extends our results in
* Bombieri-Vinogradov for multiplicative functions, and beyond the \( x^{1/2}\)-barrier * (with Fernando Shao).

(with K. Soundararajan)

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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 \( \chi\).

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

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There are many different ways to prove that there are infinitely many primes. I will highlight a few of my favourites, selected so as to involve a rich variety of mathematical ideas.