Average values of divisor sums in arithmetic progressions

Abstract

The divisor function $\tau(n)$ counts the number of positive divisors of an integer n. We are concerned with the sum $S(X,q,b)=\sum_{n \le x, n \cong b \mod q} \tau(n)$. When q=1, Drichlet derived in 1849 a pretty asymptotic formula with elementary methods. For the general case, Selberg and Hooley independently discovered the aymtotic formula $ S(X,q,b)=\frac{1}{\phi(q)}\{XP_q(\log X)+O(X^{1-\delta}\}$ for some $\de >0$ where $\phi(q)$ is the Euler phi function and $P_Q(x)$ is a linear polynomial in x. In this talk, we study the derivation of S(X,q,b) from the main term on average over b. This problem was investigated in a few papers by Banks et al. Blomer, Lu etc. We shall discuss the recent progress, applications and some ideas of proofs.