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Cross-Energy of Dirichlet Partial Sums. Part 37: The 6k±16k\pm1 race: sieve structure, Chebyshev's bias and a GRH-equivalent inequality

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.CV — Complex Variables

Contributed by Pedro Caceres ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

Every prime p>3p>3 is 6k+16k+1 or 6k−16k-1. We collect what can be proved about the two resulting sequences and add three results. The races are those modulo 33 (the prime 22 counts for the class 22); the race modulo 66 differs by that one prime. (i) In the weighting n−1/2log⁡(N/n)n^{-1/2}\log(N/n) of the Ces\`aro margin, the race between the prime powers ≡2\equiv2 and ≡1(mod3)\equiv1\pmod3 is an exact identity. The 6k−16k-1 side leads by 2L′L(12,χ−3)log⁡N−12ψ′(34)=1.1320log⁡N−1.27092\frac{L'}L(\frac12,\chi_{-3})\log N-\frac12\psi'(\frac34)=1.1320\log N-1.2709 plus the margin Mχ(N)M_\chi(N) of L(s,χ−3)L(s,\chi_{-3}). The generalized Riemann hypothesis (GRH) for L(s,χ−3)L(s,\chi_{-3}) is equivalent to Mχ≥0M_\chi\ge0 for all large NN, and under GRH the weighted race is never lost after N=2N=2. This is a criterion of the type of Hardy--Littlewood and Landau (1918) for Chebyshev's assertion, here with the Ces\`aro weight and with explicit constants. (ii) We verify 0.1024≤Mχ(N)≤0.35490.1024\le M_\chi(N)\le0.3549 for 2≤N≤2272\le N\le2^{27}, inside the window [0,0.4536][0,0.4536] that GRH predicts. We also check the formula against the first 271271 zeros of L(s,χ−3)L(s,\chi_{-3}). (iii) For the unweighted race, the bias is 1/Vχ=2.971/\sqrt{V_\chi}=2.97 standard deviations, with Vχ=∑ρ1ρ(1−ρ)=0.113230V_\chi=\sum_\rho\frac1{\rho(1-\rho)}=0.113230 (equal to ∑ρ∣ρ∣−2\sum_\rho|\rho|^{-2} under GRH); under GRH and linear independence, a recomputation with 271271 zeros gives the density 0.999060.99906, in agreement with the 0.99900.9990 of Rubinstein and Sarnak. Along the way we prove the sieve characterization of 6k±16k\pm1 and of twin primes, explain why Legendre-type counting cannot give the error term, compare twin primes and Goldbach representations modulo 66 with the Hardy--Littlewood predictions, and prove the formula for gcd⁡p(p2n−1)\gcd_p(p^{2n}-1) (the sequence 24,240,504,480,…24,240,504,480,\dots). Version~2 adds the generating function 12[ζ6(s)2±L(s,χ6)2]\frac12[\zeta_6(s)^2\pm L(s,\chi_6)^2] of the three forms 6xy±x±y6xy\pm x\pm y, and a duality theorem: averaged over the zeros of ζ\zeta, the cross-energy C2(N,12,γ)\mathcal C_2(N,\frac12,\gamma) of Part~1 returns log⁡N/N\log N/N; averaged over the zeros of L(s,χ−3)L(s,\chi_{-3}) it returns 1N∑d∣Nχ−3(d)Λ(d)\frac1N\sum_{d\mid N}\chi_{-3}(d)\Lambda(d); and M\"obius inversion separates the primes of the two lanes. Version~3 makes this exact at finite height and without RH: with a Gaussian weight on the zeros, the cross-energy satisfies an explicit formula whose pole term is C2\mathcal C_2 itself at s=0s=0, whose prime term is log⁡N\log N exactly plus a nonnegative remainder from the near coincidences nj≈Nnj\approx N, and which returns every prime power up to 6060 from the 15171517 zeros below 20002000. Nothing here proves RH, GRH, or the twin prime or Goldbach conjectures.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 25 pages, 11 figures; version 3 (2 November 2026) adds Section 11.3, an exact explicit formula for the cross-energy at finite height (unconditional), with a term-by-term explanation of how C_2 hears the primes through the zeros; version 2 added Section 11, the cross-energy as a two-way detector. Ancillary scripts in ancillary/ reproduce the computations. Part 37 of the book-series Cross-Energy of Dirichlet Partial Sums (47 parts, Parts 0-46); formerly the standalone note 'The 6k±1 race'. Renumbered 2 Nov 2026; version 2 adds Section 11. The paper states once, in its discussion, that it does not prove the Riemann hypothesis.

Provenance statement

The 6k+-1 framework and the questions are the author's and go back to his 2020 manuscript on the primes 6k+-1. The paper was written as a standalone note on 29 October 2026 and is Part 37 of the book-series Cross-Energy of Dirichlet Partial Sums; this is version 4 (10 November 2026), which recomputes the Hardy-Littlewood integrals, cites the Hardy-Littlewood-Landau line of work on Chebyshev's assertion, and changes the title. Claude (Anthropic) was used throughout as a research and writing aid: to derive and check proofs, to write and run the computations, to search the literature, and to draft text and figures. On 10 November 2026 the paper was examined twice by independent AI referee sessions that had not seen it being written; each report listed errors and recomputed the numbers it could, and this version contains the corrections from both rounds. It has not been refereed by a human reviewer. The author checked the results and is responsible for the content. The computations are floating point except where stated; the scripts are in the ancillary files (c2_regimes.py was rewritten on 10 November 2026 and reproduces its table).

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Version history

  1. v1 · 25 pagesSubmitted by Pedro CaceresRevised 10 Nov 2026 after external examination (see CHANGES_v2.md). 25 pages, 11 figures; version 3 (2 November 2026) adds Section 11.3, an exact explicit formula for the cross-energy at finite height (unconditional), with a term-by-term explanation of how C_2 hears the primes through the zeros; version 2 added Section 11, the cross-energy as a two-way detector. Ancillary scripts in ancillary/ reproduce the computations. Part 37 of the book-series Cross-Energy of Dirichlet Partial Sums (47 parts, Parts 0-46); formerly the standalone note 'The 6k±1 race'. Renumbered 2 Nov 2026; version 2 adds Section 11. The paper states once, in its discussion, that it does not prove the Riemann hypothesis.CurrentNov 11, 2026