The divisor function along arithmetic progressions

The divisor function $d(n)$ that counts the number of divisors of a given integer $n$ is one of the central functions in analytic number theory. Despite its simplicity, it is easy to formulate questions about it that are out of reach with current methods, for example when considering its distribution in arithmetic progressions.

Previous results that studied the distribution of $d(n)$ up to $X$ in arithmetic progressions of modulus $q$ covered the range $q\leq X^{2/3-\epsilon}$ as well as, on average over $q$, the range $X^{2/3+\epsilon}\leq q \leq X^{1-\epsilon}$. While it was possible to cover certain moduli in the gap $X^{2/3-\epsilon}\leq q \leq X^{2/3+\epsilon}$, these amounted to a sparse set. In this talk, I will present recent progress (joint with Jori Merikoski) in which we were able to show the expected distribution for almost all $q$ in the gap. 

Our strategy is based on assuming that $q$ has two small prime factors, Heath-Brown’s $q$-van der Corput method, and square root cancellation estimates for certain quadruple correlations of Kloosterman sums. You can expect a beginner-friendly introduction to these methods, as well as the difficulties of studying the divisor function in arithmetic progressions in general.