# The divisor function on residue classes II

Let $d(n)$ and $ct(a)$ denote the number of positive divisors of $n$ and the Ramanujan sum, respectively. The asymptotic formula for $$\sum{q\leq Q}\sum{a=1}^q\left|\sum{\substack{n\leq x\n\equiv a\pmod q}}d(n)-\frac{x}{q}\sum{t\mid q}\frac{ct(a)}{t}\left(\log\frac{x}{t^2}+2\gamma-1\right)\right|^2$$ is established for a wide range of $Q$. This generalises Motohashi's result \cite{Mot} which deals only with the special case $Q = x$ and has only a larger error term.

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size: 242 KB | mime_type: application/pdf | date: 2021-11-15

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Work Title The divisor function on residue classes II Open Access Prapanpong PongsriiamRobert C. Vaughan Article Acta Arithmetica January 1, 2018 https://doi.org/10.4064/aa161213-24-10 November 15, 2021

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