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arxiv: 2605.18996 · v1 · pith:MP2AEB6Gnew · submitted 2026-05-18 · 🧮 math.NT

The generalized Montgomery-Hooley formula: A survey

Robert C. Vaughan This is my paper

Pith reviewed 2026-05-20 07:52 UTC · model grok-4.3

classification 🧮 math.NT
keywords Montgomery-Hooley formulavariance estimatearithmetic progressionsChebyshev functionprime distributionerror terms
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The pith

The memoir surveys theorems and inequalities extending Montgomery's seminal variance estimate.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper reviews a series of theorems and inequalities that build on and broaden Montgomery's original calculation of the summed squared differences between the actual count of primes up to x in each residue class a mod q and the expected count. This calculation shows the total variance across all such classes for q up to Q is about Q times x times the log of x, with controlled error terms. A general reader might care because accurate knowledge of these variances helps explain how evenly primes are spread among different residue classes, which affects many questions about prime numbers.

Core claim

This memoir is a survey of theorems and inequalities which have grown out of, and extended, the seminal estimate of Montgomery V(x,Q) = Qx log x + O(Qx log(2x/Q)) + O(x² (log x)^{-A}).

What carries the argument

The variance V(x,Q), defined as the sum over q less than or equal to Q and a coprime to q of the squared difference |ψ(x; q, a) minus x over phi(q)|, which measures the fluctuation in prime counts across arithmetic progressions.

If this is right

  • Extensions of the formula allow for larger ranges of Q relative to x in applications to prime distribution problems.
  • Generalizations incorporate additional arithmetic functions or higher moments while preserving similar asymptotic forms.
  • The survey includes inequalities that refine the error terms under various hypotheses on the distribution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Researchers could test the formulas numerically for moderate values of x to see where the error terms become dominant.
  • Links to the distribution of primes in short intervals might be strengthened using these variance results.
  • Potential applications include improving algorithms that rely on assumptions about prime distribution in residue classes.

Load-bearing premise

The cited results from the literature, including the original Montgomery estimate and its extensions, are accurately stated and correctly interpreted in the survey.

What would settle it

A counterexample to one of the extended inequalities for specific numerical values of x and Q would indicate that the survey has included an incorrect statement.

read the original 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*}

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript is a survey compiling theorems and inequalities extending the Montgomery-Hooley variance estimate V(x,Q) = Qx log x + O(Qx log(2x/Q)) + O(x² (log x)^{-A}), originally due to Montgomery (1970). It reviews generalizations in analytic number theory, including extensions to other arithmetic functions, weighted sums, and related mean-square estimates, while contextualizing their proofs and applications.

Significance. If the cited results are reported accurately, the survey offers a useful consolidation of results in the Montgomery-Hooley tradition, aiding researchers by tracing the evolution of error terms and generalizations from the seminal estimate. No machine-checked proofs or new parameter-free derivations are claimed; the value lies in faithful contextualization of existing literature.

minor comments (2)
  1. The abstract states the original Montgomery estimate verbatim, but the survey should include a dedicated section (e.g., §2) explicitly comparing the error terms across the cited generalizations to clarify which improvements are unconditional versus conditional on RH.
  2. Notation for the variance V(x,Q) is introduced in the abstract; ensure consistent use of the same symbol and error-term conventions throughout the manuscript, particularly when discussing extensions to other functions like the von Mangoldt function.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our survey and for recommending minor revision. We will carefully review the manuscript for any minor improvements to accuracy and presentation in the revised version.

Circularity Check

0 steps flagged

No significant circularity; survey compiles external results without internal reduction

full rationale

This is a survey paper that quotes the Montgomery-Hooley estimate verbatim from the external citation HM70 and contextualizes subsequent theorems from the literature. No new derivation chain is presented, no parameters are fitted inside the manuscript, and no claim reduces by construction to a quantity defined within the survey itself. All load-bearing content consists of accurate reproduction and organization of independently established external results, satisfying the criteria for a self-contained non-circular compilation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The survey rests on standard background results in analytic number theory such as the prime number theorem in arithmetic progressions and properties of the Riemann zeta function.

axioms (1)
  • domain assumption The prime number theorem in arithmetic progressions holds with acceptable error terms for the ranges considered.
    Invoked implicitly as the foundation for the Montgomery estimate and its extensions.

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Reference graph

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