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arxiv: 2507.15334 · v2 · pith:J7DRGGWLnew · submitted 2025-07-21 · 🧮 math.NT

Refinements for primes in short arithmetic progressions

Michael Harm This is my paper

Pith reviewed 2026-05-22 00:43 UTC · model grok-4.3

classification 🧮 math.NT
keywords arithmetic progressionsprime number theoremshort intervalsDirichlet L-functionszero-density estimatesGeneralized Density Hypothesiserror terms
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The pith

Under the Generalized Density Hypothesis, the prime number theorem holds for arithmetic progressions in all intervals of length roughly sqrt(x) times exp(log x to the 2/3) and almost all intervals of length exp(log x to the 2/3).

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

The paper shows how to obtain sharper error terms in the prime number theorem for primes lying in arithmetic progressions, when the common difference is bounded by a power of the logarithm of x. It starts from a zero-free region together with an averaged zero-density estimate for the Dirichlet L-functions and derives improved bounds that let the main-term asymptotic hold inside shorter intervals than the classical ones. Under the additional Generalized Density Hypothesis the allowed lengths shrink to sqrt(x) exp((log x)^{2/3+ε}) for every such progression and to exp((log x)^{2/3+ε}) for almost every interval. Readers care because these shorter ranges give a finer picture of where primes appear inside each residue class.

Core claim

Given a zero-free region and an averaged zero-density estimate over all Dirichlet L-functions modulo q, the error terms of the prime number theorem in all and almost all short arithmetic progressions can be refined. For example, if the Generalized Density Hypothesis is assumed, then for any arithmetic progression modulo q ≤ log^ℓ x with ℓ > 0 and any ε > 0, the prime number theorem holds in all intervals (x − √x exp(log^{2/3+ε} x), x] and almost all intervals (x − exp(log^{2/3+ε} x), x] as x → ∞. This refines the classic intervals (x − x^{1/2+ε}, x] and (x − x^ε, x] for any ε > 0.

What carries the argument

Averaged zero-density estimates for the family of Dirichlet L-functions modulo q, combined with a zero-free region, to bound the remainder in the prime counting function inside short intervals.

Load-bearing premise

The existence of a zero-free region together with an averaged zero-density estimate for the Dirichlet L-functions modulo q.

What would settle it

A concrete modulus q ≤ log^ℓ x and an interval of the claimed length in which the number of primes in a given residue class differs from the main term by more than the error bound permitted under the Generalized Density Hypothesis.

read the original abstract

Given a zero-free region and an averaged zero-density estimate over all Dirichlet $L$-functions modulo $q\in\mathbb{N}$, we refine the error terms of the prime number theorem in all and almost all short arithmetic progressions. For example, if we assume the Generalized Density Hypothesis, then for any arithmetic progression modulo $q\leq \log^{\ell} x$ with $\ell>0$ and any $\varepsilon>0$, the prime number theorem holds in all intervals $(x-\sqrt{x}\exp(\log^{\frac{2}{3}+\varepsilon} x),x]$ and almost all intervals $(x-\exp(\log^{\frac{2}{3}+\varepsilon} x),x]$ as $x\rightarrow\infty$. This refines the classic intervals $(x-x^{1/2+\varepsilon},x]$ and $(x-x^\varepsilon,x]$ for any $\varepsilon>0$.

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 / 3 minor

Summary. The manuscript assumes a zero-free region and an averaged zero-density estimate over Dirichlet L-functions modulo q. Under these hypotheses, it refines the error terms in the prime number theorem for primes in arithmetic progressions with modulus q ≤ log^ℓ x. In particular, assuming the Generalized Density Hypothesis, the prime number theorem holds in all intervals (x − √x exp(log^{2/3+ε} x), x] and in almost all intervals (x − exp(log^{2/3+ε} x), x] for any ε > 0. This improves upon the classical intervals of length x^{1/2+ε} and x^ε respectively.

Significance. If the stated hypotheses hold, the work yields sharper conditional bounds on the distribution of primes in short arithmetic progressions with small moduli, obtained via standard contour integration and zero-density arguments. The averaged estimates permit a uniform treatment across all q up to a polylogarithmic range, which is a modest but useful technical improvement over pointwise estimates. The results are explicitly conditional and do not claim unconditional progress.

minor comments (3)
  1. The precise statement of the assumed zero-free region (e.g., the width in terms of q and t) should be displayed explicitly in the introduction or §2 rather than left implicit in the abstract.
  2. Notation for the averaged zero-density estimate (presumably something like an integral over |ρ| ≤ T of N(σ, T, q) or similar) should be introduced with a numbered display equation before it is invoked in the main theorems.
  3. The transition from the general hypotheses to the specific GDH case in the example could include a short remark clarifying which parameters in the zero-density estimate are optimized under GDH.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and for recommending minor revision. The report correctly captures the conditional nature of our results and the improvements obtained via averaged zero-density estimates under the Generalized Density Hypothesis.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper conditions all results on external hypotheses (zero-free region, averaged zero-density estimates over Dirichlet L-functions modulo q, and the Generalized Density Hypothesis) and derives the refined short-interval PNT statements via standard contour integration and zero-density arguments. No equation or step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation; the q ≤ log^ℓ x restriction is handled uniformly under the stated averaged estimates. The derivation is therefore self-contained against its explicit inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim depends on these standard but unproven assumptions from analytic number theory rather than new entities or fitted parameters.

axioms (3)
  • domain assumption Generalized Density Hypothesis
    This hypothesis is assumed to obtain the refined error terms for the prime number theorem in short arithmetic progressions.
  • domain assumption Existence of a zero-free region for Dirichlet L-functions
    Given as a starting point to refine the error terms.
  • domain assumption Averaged zero-density estimate over Dirichlet L-functions modulo q
    Used in conjunction with the zero-free region to derive the results.

pith-pipeline@v0.9.0 · 5669 in / 1423 out tokens · 56672 ms · 2026-05-22T00:43:53.057901+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

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Reference graph

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19 extracted references · 19 canonical work pages · cited by 2 Pith papers · 1 internal anchor

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