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 \( \chi\).

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

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

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.

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.