The generalized Montgomery-Hooley formula: A survey

Robert C. Vaughan

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arXiv · 2605.18996

The generalized Montgomery-Hooley formula: A survey

Robert C. Vaughan

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Abstract

This memoir is a survey of theorems and inequalities which have grown out of, and extended, the seminal estimate of Montgomery \cite{HM70} \begin{multline*} V(x,Q)=\sum_{q\le Q}\sum_{\substack{a=1\\ (a,q)=1}}^q \left| ψ(x;q,a) - \frac{x}{ϕ(q)} \right|^2 \\ = Qx\log x + \textstyle O\big(Qx\log\frac{2x}{Q}\big) + O\big(x^2(\log x)^{-A}\big)., \end{multline*}

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Robert C. Vaughan. 2026-05-18. The generalized Montgomery-Hooley formula: A survey. https://arxiv.org/abs/2605.18996

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