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arxiv: 2510.10853 · v2 · submitted 2025-10-12 · 🧮 math.NT

An effective Bombieri-Vinogradov error term for sifting problems

Daniel R. Johnston This is my paper

Pith reviewed 2026-05-18 07:14 UTC · model grok-4.3

classification 🧮 math.NT
keywords Bombieri-Vinogradov theoremsieve theorySiegel zeroseffective estimatestwin primessifting problemsnumber theory
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The pith

Any classical sifting problem with a Bombieri-Vinogradov style error term can be made effective without loss to its asymptotic form.

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

The paper seeks to resolve the ineffectivity that plagues many results in sieve theory due to the Bombieri-Vinogradov theorem. It demonstrates that by modifying the upper and lower bound functions in the sieve to manage Siegel-zero contributions, the same asymptotic can be achieved in an effective manner. This matters because it allows for explicit numerical bounds in problems involving the distribution of primes, such as counting twin primes. The approach applies broadly to classical sifting problems that satisfy the necessary level of distribution.

Core claim

Any classical sifting problem with a Bombieri-Vinogradov style error term can in fact be made effective, with no loss to the asymptotic form of the original ineffective result, by carefully modifying the sieve upper and lower bounds to avoid the usual complications regarding the existence of a Siegel zero.

What carries the argument

Modified upper and lower bound functions that control the Siegel-zero contribution while using a Bombieri-Vinogradov error term.

If this is right

  • One may effectively bound the number of primes p ≤ x such that p+2 is also prime by (4+o(1))C_2 x/(log x)^2.
  • Other sifting problems in additive number theory gain effective versions with the same main terms.
  • The error term remains compatible with the original level of distribution.
  • No additional losses occur in the constants or the o(1) terms compared to the ineffective case.

Where Pith is reading between the lines

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

  • This modification could lead to effective versions of other results that currently rely on ineffective Bombieri-Vinogradov applications.
  • Explicit computations of the bounds for moderate x become feasible, allowing comparison with actual prime counts.
  • Similar techniques might apply to sifting in short intervals or other variants of the sieve.

Load-bearing premise

The sifting problem admits a Bombieri-Vinogradov type theorem at a level of distribution that is compatible with the modified upper and lower bound functions introduced to control the Siegel-zero contribution.

What would settle it

A computation showing that the effective twin prime bound fails to hold asymptotically, for example if the count exceeds the predicted (4+o(1))C_2 x/(log x)^2 by more than the allowed error for large x.

read the original abstract

In number theory, many major results related to the additive properties of primes are proven using the methods of sieve theory. However, in nearly every case, the existing proofs of these results are ineffective, in that explicit values for which they hold cannot be computed. The reason for this ineffectivity is due to the reliance on the Bombieri--Vinogradov theorem. In this paper, we show that any classical sifting problem with a Bombieri--Vinogradov style error term can in fact be made effective, with no loss to the asymptotic form of the original (ineffective) result. This is done by carefully modifying the sieve upper and lower bounds as to avoid the usual complications regarding the existence of a Siegel zero. We also provide some simple applications. For example, we show that one may effectively bound the number of primes $p\leq x$ such that $p+2$ is also prime by \begin{equation*}(4+o(1))C_2\frac{x}{(\log x)^2},\end{equation*}where\begin{equation*}C_2=2\prod_{p>2}\left(1-\frac{1}{(p-1)^2}\right).\end{equation*}

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

2 major / 2 minor

Summary. The paper claims that any classical sifting problem admitting a Bombieri-Vinogradov style error term can be made effective, with no loss to the asymptotic form, by redefining the sieve upper and lower bound functions to absorb the possible Siegel-zero contribution. This is achieved via a general construction, with applications including an effective bound on the number of primes p ≤ x such that p+2 is also prime, given by (4+o(1))C_2 x/(log x)^2 where C_2 is the twin-prime constant.

Significance. If the central claim holds, the result is significant for analytic number theory because it converts a broad class of currently ineffective sieve results into effective ones while preserving the main asymptotic term. This addresses a persistent obstacle arising from possible exceptional zeros and could enable explicit versions of many prime-tuple and additive prime problems. The generality of the method and the concrete applications (such as the twin-prime bound) add practical value.

major comments (2)
  1. [Main Theorem] Main Theorem statement: the construction of the modified upper and lower sifting functions must be shown to preserve a Bombieri-Vinogradov theorem at precisely the same level of distribution as the original problem. If the modification enlarges the support or changes the multiplicative structure, the required level could increase, undermining applicability to classical problems already at the boundary of known distribution ranges (e.g., x^{1/2-ε}). Explicit verification of the level in the error-term estimates is needed.
  2. [Error estimates section] Error-term bookkeeping (following the redefinition of weights): the o(1) terms after absorbing the Siegel-zero contribution must remain uniform and independent of any exceptional zero; the current presentation leaves the proportionality of these errors implicit.
minor comments (2)
  1. [Abstract] Abstract: the displayed twin-prime bound is clear, but the constant C_2 should be cross-referenced to its definition in the general theorem for consistency.
  2. [Introduction] Notation: ensure that the modified functions β_d and β'_d are introduced with uniform notation throughout the proofs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. The two major comments identify places where additional explicit verification and bookkeeping would strengthen the presentation. We address each below and will incorporate the necessary clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Main Theorem] Main Theorem statement: the construction of the modified upper and lower sifting functions must be shown to preserve a Bombieri-Vinogradov theorem at precisely the same level of distribution as the original problem. If the modification enlarges the support or changes the multiplicative structure, the required level could increase, undermining applicability to classical problems already at the boundary of known distribution ranges (e.g., x^{1/2-ε}). Explicit verification of the level in the error-term estimates is needed.

    Authors: We agree that explicit verification of the preserved level of distribution is required for the result to apply at the boundary of known ranges. The modified weights are obtained by a multiplicative adjustment that factors out the possible Siegel-zero contribution while leaving the support and the underlying multiplicative structure unchanged up to a factor of 1 + O(1/log log x). In the revision we will insert a short lemma immediately after the definition of the modified sifting functions that recomputes the Bombieri-Vinogradov error term for the new weights and confirms that the same level of distribution is retained. This will make the applicability to problems such as the twin-prime problem at level x^{1/2-ε} fully transparent. revision: yes

  2. Referee: [Error estimates section] Error-term bookkeeping (following the redefinition of weights): the o(1) terms after absorbing the Siegel-zero contribution must remain uniform and independent of any exceptional zero; the current presentation leaves the proportionality of these errors implicit.

    Authors: The referee is correct that the uniformity of the remaining o(1) terms should be stated explicitly. These terms originate from the original sieve estimates and the Bombieri-Vinogradov theorem, both of which are already uniform with respect to any exceptional zero once the Siegel contribution has been absorbed into the modified bounds. In the revised version we will add a short paragraph in the error-estimates section that records this uniformity and makes the implied constants independent of any possible Siegel zero explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external BV assumption and standard identities

full rationale

The paper takes as given a Bombieri-Vinogradov error term at a compatible level for the original sifting problem, then modifies the upper and lower bound functions to neutralize Siegel-zero effects while preserving the asymptotic form. This modification is constructed explicitly from the standard sieve weights and the assumed error term; the target effective bound is not obtained by fitting any parameter inside the paper nor by redefining the input in terms of the output. No load-bearing step reduces to a self-citation chain, an imported uniqueness theorem, or an ansatz smuggled via prior work by the same author. The argument therefore remains self-contained against the stated external hypothesis and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument rests on the standard analytic properties of sieve functions and the existence of a Bombieri-Vinogradov theorem at a suitable level; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The sifting problem satisfies a Bombieri-Vinogradov type theorem with level of distribution D = x^theta for some theta > 0
    Invoked to supply the error term that is then made effective by the new bound modification.

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