The divisor function on residue classes II
Volume 182 / 2018
Acta Arithmetica 182 (2018), 133-181
MSC: Primary 11N37; Secondary 11A25, 11B25.
DOI: 10.4064/aa161213-24-10
Published online: 22 January 2018
Abstract
Let $d(n)$ and $c_t(a)$ denote the number of positive divisors of $n$ and the Ramanujan sum, respectively. The asymptotic formula for $$ \sum_{q\leq Q}\sum_{a=1}^q\biggl|\sum_{\substack{n\leq x\\n\equiv a\,({\rm mod}\, q)}} d(n)-\frac{x}{q}\sum_{t\mid q}\frac{c_t(a)}{t}\biggl(\log\frac{x}{t^2}+2\gamma-1\biggr)\biggr|^2 $$ is established for a wide range of $Q$. This generalises Motohashi’s 1973 result which deals only with the special case $Q = x$ and has a larger error term.
