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Making the coprimality convention explicit in a mean value theorem of Halupczok for the ternary Goldbach problem

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Steve Hubbard ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

Theorem 1.2 of Halupczok (2013) is a Bombieri-Vinogradov type mean value theorem for the ternary Goldbach problem with one prime in an arithmetic progression and two primes in short intervals. Her proof works with residue classes a mod qa \bmod q coprime to the modulus, as stated on p. 344 of her paper, and this convention needs to be written explicitly into the statement because the printed product and the prime sum differ at a=q=3a = q = 3, where the prime sum is 00 for X1>3X_1 > 3 while the printed product is at least 9/169/16 for odd nn. We give two closely related explicit formulations, Formulation A and Formulation B, show that Formulation A implies Formulation B, and prove both assuming two cited inputs, Hypothesis H1 and Hypothesis H2, or alternatively Theorems 2.1 and 2.3 of her paper, which follow from these two inputs. Here Hypothesis H1 is a uniform reading of a printed remark of Perelli, Pintz and Salerno, supported by a reading of the proof, and Hypothesis H2 is assembled from displays of Perelli and Pintz and is not a numbered statement. The supplement of Theorem 1.2 with a maximum over aa is not covered. The later results of Halupczok's paper use only coprime residue classes and are unaffected. The main results of this note have been checked by computer.

Provenance statement

The text was drafted by AI models, Claude and Gemini, under the direction of Steve Hubbard; the main results are checked in Lean 4 with Mathlib, source at github.com/sthubbar/goldbach-lean. Agent session logs are not attached because they contain private email.

References

  1. K. Halupczok, Goldbach's problem with primes in arithmetic progressions and in short intervals , J. Théor.\ Nombres Bordeaux 25 (2013), no. 2, 331 to 351, doi:10.5802/jtnb.839. Pages used: Theorem 1.2 pp. 333 to 334; Theorem 2.1 p. 338; Theorem 2.3 p. 339; displays (1.1) and (1.2) p. 335; the proof sentence on (q,a)=1(q, a) = 1 p. 344.DOI
  2. K. Halupczok, Goldbach's problem with primes in arithmetic progressions and in short intervals , preprint, arXiv:1212.4406, v1, 18 Dec 2012.arXiv
  3. A. Perelli, J. Pintz and S. Salerno, Bombieri's theorem in short intervals , Ann.\ Scuola Norm.\ Sup.\ Pisa Cl.\ Sci.\ (4) 11 (1984), no. 4, 529 to 539, numdam ASNSP_1984_4_11_4_529_0. Pages used: display (2) p. 529; THEOREM and REMARK p. 530; proof pp. 531 to 538, display (5) p. 531.
  4. A. Perelli and J. Pintz, On the exceptional set for Goldbach's problem in short intervals , J. London Math.\ Soc.\ (2) 47 (1993), 41 to 49, doi:10.1112/jlms/s2-47.1.41. Pages used: introduction p. 41; R∗R^* and M∗M^* p. 43; displays (1), (3), (4), (5) and Theorem 3 p. 44; §4 p. 45; §5 p. 46.DOI
  5. M. N. Huxley and H. Iwaniec, Bombieri's theorem in short intervals , Mathematika 22 (1975), 188 to 194.
  6. H. A. Helfgott, Major arcs for Goldbach's problem , preprint, arXiv:1305.2897, 2013; see also Minor arcs for Goldbach's problem , preprint, arXiv:1205.5252, 2012.arXiv
  7. Palomar registry entry PALOMAR-2026-10-28-000026 , version 1, https://palomar-registry.org/entry.html?id=PALOMAR-2026-10-28-000026&version=1.link
  8. S. Hubbard, goldbach-lean , https://github.com/sthubbar/goldbach-lean, entries/halupczok2012 , commit 12f09bfeca2ae00988fa039e2e82c6de4d147108 .link
  9. PrimeNumberTheoremAnd , https://github.com/AlexKontorovich/PrimeNumberTheoremAnd.link
  10. The mathlib Community, The Lean mathematical library , https://github.com/leanprover-community/mathlib4, version v4.35.0-rc3 .link

Version history

  1. v1 · 8 pagesSubmitted by Steve HubbardInitial depositCurrentNov 08, 2026