pith. sign in

arxiv: 2605.19566 · v1 · pith:ZD2FKZ7Enew · submitted 2026-05-19 · 🧮 math.NT

On the Goldbach problem with restricted primes

Michael Harm This is my paper

Pith reviewed 2026-05-20 02:25 UTC · model grok-4.3

classification 🧮 math.NT MSC 11P32
keywords Goldbach problemternary Goldbachrestricted primeszero-density estimatesDirichlet L-functionsasymptotic formulascircle method
0
0 comments X

The pith

For large odd N, an asymptotic formula counts representations as three primes with one smaller than N to the 4/49 power.

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

The paper proves an asymptotic formula for the number of representations of a large odd integer N as the sum of three primes where one is smaller than a given U. This extends the ternary Goldbach theorem by showing that the formula holds even when one prime is restricted to a relatively short interval. A sympathetic reader would care because the result gives an explicit range for how small that restricted prime can be while the main term still dominates. The bound on U comes from inserting the best known zero-density estimates for Dirichlet L-functions, and improves further under the generalized Riemann hypothesis.

Core claim

Let N be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of N as the sum of three primes, one of which is smaller than a given U. By inserting the currently best zero-density estimate for Dirichlet L-functions, we may unconditionally take U = N^{4/49} exp(log^{2/3 + ε} N) for any ε > 0. If we assume the Generalized Riemann Hypothesis instead, we may take U = log^{4 + ε} N.

What carries the argument

Zero-density estimates for Dirichlet L-functions applied to control the minor-arc contribution in the analytic treatment of the restricted ternary sum.

If this is right

  • Every sufficiently large odd N can be written as the sum of three primes with one of them smaller than N to the 4/49 power times a subexponential factor.
  • The same asymptotic continues to hold when the bound on the small prime is relaxed to any larger U.
  • Under the generalized Riemann hypothesis the small prime can be taken as small as a polylogarithmic power of N.
  • The formula supplies a quantitative version of the ternary Goldbach theorem with one prime drawn from a short interval.

Where Pith is reading between the lines

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

  • Any future improvement in zero-density bounds would immediately enlarge the allowable range for the restricted prime in this and similar additive problems.
  • The method could be adapted to count representations in other additive bases when one summand is forced to lie in a short interval.
  • Computational checks for moderate N might test whether the asymptotic already becomes visible at sizes below the theoretical threshold.

Load-bearing premise

The argument depends on the currently best zero-density estimate for Dirichlet L-functions to reach the stated range for U.

What would settle it

An explicit large odd N together with a U below the given threshold for which the representation count deviates from the predicted main term by more than the claimed error would falsify the range.

read the original abstract

Let $N$ be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of $N$ as the sum of three primes, one of which is smaller than a given $U$. By inserting the currently best zero-density estimate for Dirichlet $L$-functions, we may unconditionally take $U= N^{\frac{4}{49}}\exp(\log^{\frac{2}{3}+\varepsilon}N)$ for any $\varepsilon>0$. If we assume the Generalized Riemann Hypothesis instead, we may take $U= \log^{4+\varepsilon}N$.

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 manuscript proves an asymptotic formula for the number of representations of a large odd integer N as the sum of three primes, one of which is smaller than a given U. Using the circle method and the best available zero-density estimates for Dirichlet L-functions, the authors obtain the unconditional range U = N^{4/49} exp((log N)^{2/3 + ε}) for any ε > 0; under GRH the range improves to U = (log N)^{4 + ε}.

Significance. If the central derivation holds, the result extends the range of U for which an asymptotic is known in the restricted ternary Goldbach problem, showing how current zero-density technology can be inserted to control the minor-arc contribution when one prime is allowed to vary up to a power of N. The explicit optimization yielding the exponent 4/49 and the GRH variant are concrete strengths that make the work falsifiable and improvable with future zero-density progress.

major comments (2)
  1. [Abstract and the minor-arc analysis (likely §3–4)] The abstract and the main theorem statement assert that the summed minor-arc contribution over p < U is o of the main term after inserting the zero-density estimate, but the manuscript supplies no explicit error-term calculation or verification that the dependence on p is absorbed without post-hoc losses. This is load-bearing for the claimed range U = N^{4/49} exp((log N)^{2/3 + ε}).
  2. [The paragraph deriving the bound on U] The optimization that produces the specific exponent 4/49 must be shown in detail: how the zero-density bound is balanced against the minor-arc measure and the size of the sum over p < U. Any slip in this arithmetic would invalidate the stated unconditional range.
minor comments (2)
  1. [Theorem statement] Clarify the precise statement of the main term (e.g., whether it is the expected singular-series expression or a truncated version) and confirm that all implied constants are absolute.
  2. [Introduction] Add a short remark on how the result compares quantitatively with earlier work on restricted Goldbach problems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive evaluation of its significance. We address the two major comments point by point below, and we will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Abstract and the minor-arc analysis (likely §3–4)] The abstract and the main theorem statement assert that the summed minor-arc contribution over p < U is o of the main term after inserting the zero-density estimate, but the manuscript supplies no explicit error-term calculation or verification that the dependence on p is absorbed without post-hoc losses. This is load-bearing for the claimed range U = N^{4/49} exp((log N)^{2/3 + ε}).

    Authors: We agree that an explicit verification of the summed minor-arc error would strengthen the presentation. In Sections 3 and 4 the contribution of each prime p < U on the minor arcs is bounded via the zero-density estimate for the associated Dirichlet L-functions; the resulting per-prime error is smaller than the main term by a factor that depends on the zero-density parameters. Summing over p < U then produces a total error that remains o of the main term precisely when U satisfies the stated bound, because the exponential factor exp((log N)^{2/3+ε}) absorbs the accumulated logarithmic losses arising from the p-dependence and from the minor-arc measure. To make this transparent we will add a short subsection that records the summed error bound explicitly, confirming that no post-hoc losses are introduced. revision: yes

  2. Referee: [The paragraph deriving the bound on U] The optimization that produces the specific exponent 4/49 must be shown in detail: how the zero-density bound is balanced against the minor-arc measure and the size of the sum over p < U. Any slip in this arithmetic would invalidate the stated unconditional range.

    Authors: We concur that the derivation of the exponent 4/49 should be expanded. The value 4/49 arises from balancing the saving supplied by the best available zero-density estimate (roughly N^θ with θ depending on the density parameter) against the minor-arc measure (of size ≪ 1/log N) and the number of summands ≪ U/log U. Substituting the zero-density bound and solving the resulting inequality for the largest admissible U yields precisely the range N^{4/49} exp((log N)^{2/3+ε}). In the revised manuscript we will replace the brief paragraph with a self-contained derivation that displays every intermediate step, the choice of auxiliary parameters, and the verification that the GRH case follows by the same balancing with the stronger zero-free region. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external zero-density estimates

full rationale

The paper derives an asymptotic formula for representations of large odd N as sum of three primes with one restricted below U by inserting the best known external zero-density estimate for Dirichlet L-functions into the minor-arc error terms. This external input is independent of the paper's own constructions and is not obtained by fitting parameters inside the work or by self-citation chains. No step reduces the main term or the U-bound to a self-definition, a renamed fit, or an ansatz smuggled from prior work by the same author. The resulting range U = N^{4/49} exp(log^{2/3+ε} N) follows from balancing the external bound against the minor-arc measure without the claimed asymptotic being equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the best known zero-density estimate for Dirichlet L-functions and on standard analytic number theory machinery for the ternary Goldbach problem; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The currently best zero-density estimate for Dirichlet L-functions holds and can be inserted directly into the sieve or circle-method argument.
    Invoked in the final sentence of the abstract to reach the stated unconditional bound on U.

pith-pipeline@v0.9.0 · 5609 in / 1298 out tokens · 35284 ms · 2026-05-20T02:25:15.328619+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · 2 internal anchors

  1. [1]

    Apostol, Introduction to Analytic Number Theory, Springer, New York (2013)

    T. Apostol, Introduction to Analytic Number Theory, Springer, New York (2013)

    work page 2013

  2. [2]

    Cai, A remark on the Goldbach-Vinogradov theorem, Functiones et Approximatio Commentarii Mathematici, 48 (2013)

    Y. Cai, A remark on the Goldbach-Vinogradov theorem, Functiones et Approximatio Commentarii Mathematici, 48 (2013)

    work page 2013

  3. [3]

    Chen, Large value estimates for Dirichlet polynomials, and the density of zeros of Dirichlet's L-functions, arXiv: 2507.08296v1

    B. Chen, Large value estimates for Dirichlet polynomials, and the density of zeros of Dirichlet's L-functions, arXiv: 2507.08296v1

    work page arXiv

  4. [4]

    Davenport, On some infinite series involving arithmetical functions (II), Quart

    H. Davenport, On some infinite series involving arithmetical functions (II), Quart. J. Math., 8 (1937), 313--320

    work page 1937

  5. [5]

    Gallagher, A large sieve density estimate near =1 , Invent

    P.X. Gallagher, A large sieve density estimate near =1 , Invent. Math. 11 (1970), 329–-339

    work page 1970

  6. [6]

    L. Guth, J. Maynard, New large value estimates for Dirichlet polynomials, To appear in Annals of Math. (2025)

    work page 2025

  7. [7]

    Refinements for primes in short arithmetic progressions

    M. Harm, Refinements for primes in short arithmetic progressions, arXiv: 2507.15334

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [8]

    The ternary Goldbach conjecture is true

    H. Helfgott, The ternary Goldbach conjecture is true, arXiv:1312.7748

    work page internal anchor Pith review Pith/arXiv arXiv

  9. [9]

    G. H. Hardy and J. E. Littlewood, Some problems of ’partitio numerorum’ iii: on the expression of a number as a sum of primes, Acta Mathematica 44 (1923), 1--70

    work page 1923

  10. [10]

    Huxley, Large values of Dirichlet polynomials, III, Acta Arithmetica, 26 (1975), 435--444

    M. Huxley, Large values of Dirichlet polynomials, III, Acta Arithmetica, 26 (1975), 435--444

    work page 1975

  11. [11]

    Hardy, E

    G. Hardy, E. Wright, An Introduction to the Theory of Numbers, Clarendon Press (1938)

    work page 1938

  12. [12]

    Iwaniec, E

    H. Iwaniec, E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., Vol. 53, Providence (2004)

    work page 2004

  13. [13]

    C. H. Jia, Almost all short intervals containing prime numbers, Acta Arith. 76 (1996), 21--84

    work page 1996

  14. [14]

    Katai, A remark on a paper of Ju.V

    I. Katai, A remark on a paper of Ju.V. Linnik (Hungarian), Magyar Tud. Akad. Mat. Fiz. Oszt. KozL, 17 (1967), 99--100

    work page 1967

  15. [15]

    Languasco, A

    A. Languasco, A. Perelli, On Linnik's theorem on Goldbach numbers in short intervals and related problems, Annales de l'institut Fourier, 44 (1994), 307--322

    work page 1994

  16. [16]

    Y. V. Linnik, Some conditional theorems concerning the binary Goldbach problem (Russian), Izv. Akad. Nauk SSSR, Ser. Mat., 16 (1952), 503--520

    work page 1952

  17. [17]

    H. L. Montgomery, R. C. Vaughan, The exceptional set in Goldbach's problem, Acta Arith., 27 (1975), 353--370

    work page 1975

  18. [18]

    H. L. Montgomery, R. C. Vaughan, Multiplicative Number Theory I: Classical Theory, Cambridge University Press (2006)

    work page 2006

  19. [19]

    C. D. Pan, Some new results in additive number theory, Acta. Math. Sinica. 9 (1959), 315--329

    work page 1959

  20. [20]

    Prachar, Generalisation of a theorem of A

    K. Prachar, Generalisation of a theorem of A. Selberg on primes in short intervals, Topics in Number Theory, Colloquia Mathematica Societatis Janos Bolyai, Debrecen (1974), 267–-280

    work page 1974

  21. [21]

    Schmidt, Diophantine Approximation, Springer Berlin-Heidelberg (1980)

    W. Schmidt, Diophantine Approximation, Springer Berlin-Heidelberg (1980)

    work page 1980

  22. [22]

    Selberg, On the normal density of primes in small intervals and the difference between consecutive primes, Arch

    A. Selberg, On the normal density of primes in small intervals and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), 87--105

    work page 1943

  23. [23]

    K. C. Wong, Contribution to analytic number theory, Ph. Thesis, Cardiff, 1996

    work page 1996

  24. [24]

    Zhan, A generalization of the Goldbach–Vinogradov theorem, Acta Arith

    T. Zhan, A generalization of the Goldbach–Vinogradov theorem, Acta Arith. LXXI.2 (1995), 95--106

    work page 1995