The generalized Montgomery-Hooley formula: A survey

TL;DR

This survey reviews the development and extensions of the Montgomery-Hooley formula, highlighting its significance in understanding prime number distributions and related inequalities.

Contribution

It compiles and discusses various theorems and inequalities derived from the original Montgomery estimate, emphasizing their advancements and applications.

Findings

Extended the Montgomery-Hooley formula to broader contexts

Summarized key inequalities related to prime distributions

Highlighted the formula's impact on analytic number theory

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| \psi(x;q,a) - \frac{x}{\phi(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*}