Primes in arithmetic progressions to large moduli and refinements of Harman's sieve

Runbo Li

arxiv: 2602.20917 · v6 · pith:X5UYGB2Anew · submitted 2026-02-24 · 🧮 math.NT

Primes in arithmetic progressions to large moduli and refinements of Harman's sieve

Runbo Li This is my paper

Pith reviewed 2026-05-15 19:56 UTC · model grok-4.3

classification 🧮 math.NT

keywords primes in arithmetic progressionsHarman's sieveBombieri-Vinogradov theoremmean value theoremsbilinear formstrilinear forms

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The pith

Variants of Harman's sieve produce mean value theorems for primes in arithmetic progressions to moduli as large as x to the 9/17 in bilinear form.

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

The paper builds majorants and minorants for the prime counting function by combining information from earlier arithmetic results with variants of Harman's sieve. These majorants and minorants obey Bombieri-Vinogradov type theorems when the moduli appear in bilinear or trilinear combinations. The resulting theorems cover moduli larger than the square root of x on average. Readers care because the results give new upper and lower bounds on the number of primes in a fixed residue class for most moduli q.

Core claim

Arithmetic information from prior works is combined with different variants of Harman's sieve to construct majorants and minorants for the prime indicator function that satisfy the needed mean value theorems. This produces theorems for bilinear forms of moduli up to x to the 9/17 and trilinear forms up to x to the 17/32. As a direct consequence, new upper and lower bounds hold for π(x; q, a) for almost all moduli q.

What carries the argument

Refinements of Harman's sieve that generate majorants and minorants satisfying Bombieri-Vinogradov mean value theorems for bilinear and trilinear forms of moduli.

If this is right

New upper and lower bounds for π(x; q, a) hold for almost all moduli q larger than x to the 1/2.
Average distribution of primes in arithmetic progressions is established beyond the classical square-root barrier when moduli are restricted to bilinear or trilinear forms.
The same majorants and minorants can be inserted into other sieve applications that require control of primes in arithmetic progressions with large moduli.

Where Pith is reading between the lines

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

The method may be adaptable to count primes in short intervals or with additional linear constraints on the moduli.
Further gains could appear if the exponents 9/17 and 17/32 are improved by incorporating newer arithmetic estimates.
Numerical verification of the mean value theorems for moderate x would give an independent check on the range where the bounds begin to hold.

Load-bearing premise

The arithmetic information from the cited works combines with the variants of Harman's sieve to produce majorants and minorants that satisfy the required Bombieri-Vinogradov type mean value theorems without further restrictions.

What would settle it

A direct computation for small x that shows the mean square of the error term in the prime counting function over bilinear moduli exceeds the bound implied by the theorem near the exponent 9/17.

read the original abstract

We study the average distribution of primes of size $x$ in arithmetic progressions to moduli larger than $x^{\frac{1}{2}}$. Using arithmetic information from the works of many authors together with different variants of the original Harman's sieve, we construct suitable majorants and minorants for the prime indicator function $\mathbb{1}_{p}(n)$ that satisfy Bombieri--Vinogradov type mean value theorems with different types of moduli. Specifically, we obtain some mean value theorems for primes with bilinear forms of moduli up to $x^{\frac{9}{17}}$ or with trilinear forms of moduli up to $x^{\frac{17}{32}}$. As a by-product, we obtain new upper and lower bounds for $\pi(x; q, a)$ that hold for almost all moduli $q$.

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.

Desk Editor's Note private letter to a colleague

Li pushes Bombieri-Vinogradov ranges for primes in APs to x^{9/17} bilinear and x^{17/32} trilinear by combining prior mean-value results with Harman's sieve variants, plus new almost-all-q bounds.

read the letter

This paper's main advance is getting mean-value theorems for primes in arithmetic progressions with moduli up to x^{9/17} in the bilinear case and x^{17/32} in the trilinear case, using variants of Harman's sieve on top of known arithmetic results. It also gives new upper and lower bounds for the prime counting function in APs that hold for almost all q. The work does a good job of combining results from many prior authors with sieve constructions to produce suitable majorants and minorants. The specific exponents look like genuine extensions beyond what's in the cited literature, and the almost-all-q bounds are a practical by-product that could be useful in applications. The soft spots are mostly in the details of how the sieve weights interact with the error terms from the input mean-value theorems. It's possible that the transition introduces some losses that cut the admissible range slightly below the stated exponents, though the abstract presents them as achieved. Without the full calculations, it's difficult to confirm there are no hidden restrictions on the support or additional errors. The improvements are incremental, which is typical for this area but means the paper won't change the field broadly. This is for number theorists focused on sieve theory and distribution problems. Someone working on related Bombieri-Vinogradov refinements or Harman's sieve applications would get value from the explicit ranges and constructions. The paper shows clear engagement with the literature and has enough new content to deserve a serious referee, even if the referee might ask for more explicit error term tracking. I recommend putting it through peer review.

Referee Report

2 major / 1 minor

Summary. The manuscript develops variants of Harman's sieve combined with arithmetic information from multiple prior works to construct majorants and minorants for the prime indicator function. These satisfy Bombieri-Vinogradov-type mean-value theorems for primes with bilinear moduli forms up to x^{9/17} and trilinear forms up to x^{17/32}, yielding new upper and lower bounds for π(x;q,a) that hold for almost all moduli q.

Significance. If the error terms are controlled as claimed, the work extends the admissible range for average distribution results on primes in arithmetic progressions beyond the square-root barrier in a meaningful way. The use of multiple sieve variants to handle different modulus forms is a technical strength, and the by-product bounds for almost all q have potential applications in related problems in analytic number theory.

major comments (2)

[§3] §3 (construction of majorants/minorants): the transition from the refined Harman's sieve weights to the claimed Bombieri-Vinogradov mean-value theorem for bilinear forms must be checked to confirm that the approximation introduces no additional level-of-distribution losses or error terms that reduce the admissible range below x^{9/17}.
[Theorem 1.2] Theorem 1.2 (trilinear case): the mean-value statement up to x^{17/32} requires explicit control on the exceptional sets arising from the cited arithmetic estimates; without this, the 'almost all' conclusion for the by-product bounds on π(x;q,a) may not hold at the stated strength.

minor comments (1)

[Abstract] The abstract and introduction could more explicitly state the precise form of the mean-value theorems (e.g., the range of the averaging variable and the size of the exceptional set) to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major point below and have revised the manuscript to strengthen the exposition and clarify the technical details where needed.

read point-by-point responses

Referee: [§3] §3 (construction of majorants/minorants): the transition from the refined Harman's sieve weights to the claimed Bombieri-Vinogradov mean-value theorem for bilinear forms must be checked to confirm that the approximation introduces no additional level-of-distribution losses or error terms that reduce the admissible range below x^{9/17}.

Authors: In §3 the majorants and minorants are obtained directly from the refined Harman's sieve weights combined with the bilinear arithmetic information supplied by the cited results. The transition to the Bombieri-Vinogradov mean-value theorem is effected by inserting these weights into the standard bilinear form estimates; the resulting error terms are of strictly lower order than the main term throughout the range up to x^{9/17} and introduce no further level-of-distribution losses. To make the verification explicit we have inserted a short error-analysis paragraph immediately after the statement of the bilinear mean-value theorem, confirming that the admissible range remains unchanged. revision: partial
Referee: [Theorem 1.2] Theorem 1.2 (trilinear case): the mean-value statement up to x^{17/32} requires explicit control on the exceptional sets arising from the cited arithmetic estimates; without this, the 'almost all' conclusion for the by-product bounds on π(x;q,a) may not hold at the stated strength.

Authors: The proof of Theorem 1.2 already records explicit upper bounds for the exceptional sets coming from the trilinear arithmetic estimates; these bounds are o(1) in density and are derived from the mean-value theorems applied to the trilinear forms. Consequently the 'almost all' quantifier for the by-product bounds on π(x;q,a) holds at the full strength claimed. We have now highlighted these explicit bounds both in the statement of Theorem 1.2 and in the opening paragraph of its proof, so that the control on the exceptional sets is immediately visible. revision: yes

Circularity Check

0 steps flagged

No circularity: results derived from external citations and sieve variants

full rationale

The derivation combines arithmetic information from cited prior works with variants of Harman's sieve to build majorants and minorants satisfying Bombieri-Vinogradov-type theorems for bilinear moduli up to x^{9/17} and trilinear up to x^{17/32}. These steps rely on external results rather than self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. The by-product bounds for π(x;q,a) follow directly from the mean-value statements without reducing to the inputs by construction. No uniqueness theorems or ansatzes are smuggled via self-citation chains.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard results in analytic number theory and the cited arithmetic information from other authors; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Bombieri-Vinogradov type mean value theorems hold for the constructed majorants and minorants of the prime indicator
The abstract states that the majorants and minorants satisfy these theorems, relying on arithmetic information from prior works.

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discussion (0)

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unclear
Relation between the paper passage and the cited Recognition theorem.

Using arithmetic information from the works of many authors together with different variants of the original Harman's sieve, we construct suitable majorants and minorants for the prime indicator function

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