The divisor function on residue classes III

Abstract

Let $n, q, a\in \mathbb N$, d(n) the number of positive divisors of n, and A(x, q, a) the sum of d(n) over $n\le x$ and $n\equiv a\pmod q$. Previously we obtained an asymptotic formula for the second moment

$$\begin{aligned} V(x,Q) = \sum _{q\le Q}\sum _{a=1}^q \left| A(x,q,a)-M(x,q,a)\right| ^2, \end{aligned}$$

where $Q\le x$ and M(x, q, a) is a usual expected value for A(x, q, a). In this article, we compute the second moment similar to the previous one but M(x, q, a) is replaced by a better approximating function $\rho _R(x,q,a)$ given by

$$\begin{aligned} \rho _R(x,q,a) = \sum _{\begin{array}{c} n\le x\\ n\equiv a(\bmod \, q) \end{array}}\sum _{r\le R}\frac{c_r(n)}{r}\left( \log \frac{n}{r^2}+2\gamma \right) , \end{aligned}$$

where $c_r(n)$ is Ramanujan’s sum. The advantage of using $\rho _R(x,q,a)$ instead of M(x, q, a) is that the new second moment, for a suitable R, gives us practically the same main term and the error term of smaller or the same order of magnitude with much simpler and shorter calculation than that of V(x, Q).