Arithmetic exponent pairs for algebraic trace functions and applications

Abstract

We study short sums of algebraic trace functions via the $q$-analogue of the van der Corput method, and develop theory of arithmetic exponent pairs that coincide with the classical case when the moduli have sufficiently good factorizations. As an application, we prove a quadratic analogue of the Brun–Titchmarsh theorem on average, bounding the number of primes $p⩽X$ such that $p^2+1\equiv 0\text{mod}q$. The other two applications include a larger level of distribution of divisor functions in arithmetic progressions and a sub-Weyl subconvex bound of Dirichlet $L$-functions studied previously by Irving.