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3:00pm, Wednesday 9 July

Abstract

Given a zero-free region and an average zero-density estimate for all Dirichlet $L$-functions modulo $q$, we refine the error terms of the prime number theorem in all and almost all short arithmetic progressions. If we e.g. assume the Generalized Density Hypothesis, then as $x\rightarrow \infty$ the prime number theorem holds for any arithmetic progression modulo $q\leq \log^\ell x$  for any $\ell>0$ and in the interval $(x,x+\sqrt{x}\exp(\log^{2/3+\varepsilon} x)]$ for any $\varepsilon>0$. This refines the classic interval $(x,x+x^{1/2+\varepsilon}]$.