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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Work Title The divisor function on residue classes II
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Open Access
Creators
  1. Prapanpong Pongsriiam
  2. Robert C. Vaughan
License In Copyright (Rights Reserved)
Work Type Article
Publisher
  1. Acta Arithmetica
Publication Date January 1, 2018
Publisher Identifier (DOI)
  1. https://doi.org/10.4064/aa161213-24-10
Deposited November 15, 2021

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