HEXAGONbeta

Cross-Energy of Dirichlet Partial Sums. Part 38: The energy of the Chebyshev race: an explicit unconditional floor for the prime race between the lanes 6k±16k\pm1

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

Let R(x)=π(x;3,2)−π(x;3,1)R(x)=\pi(x;3,2)-\pi(x;3,1) be the Chebyshev race between the two prime lanes 6k∓16k\mp1 (the prime 22 counted, by the congruence 2≡−12\equiv-1 mod 33, in the 6k−16k-1 lane). We study the race as an \emph{energy} and establish three things (the energy law uses two estimates imported from the ζ\zeta case; see Proposition~\ref{prop:cramer}). First, an elementary ledger identity: the entire bias of the race is the bookkeeping of prime squares, which all land in the lane 6k+16k+1; subtracting the ledger 12π(x)\tfrac12\pi(\sqrt x) turns a walk with \emph{zero} sign changes up to 10810^8 into one with 15,32915{,}329 (21,32921{,}329 if every passage between D>0D>0 and D≤0D\le0 at a prime is counted). Second, the mean energy of the normalized race signal e(u)=e−u/2ψ(eu,χ3)e(u)=e^{-u/2}\psi(e^u,\chi_3) admits an explicit unconditional lower bound for its upper limit, which we call the floor, \[ \limsup_{U\to\infty}\frac1U\int_0^U e(u)^2\,du\;\ge\;\kappa_3 :=\sum_{\rho}\frac{1}{\rho(1-\rho)} =\log\frac3\pi-\gamma_E+2\,\frac{L'}{L}(1,\chi_3) =0.113229969857\ldots, \] the sum running over the nontrivial zeros of L(s,χ3)L(s,\chi_3) with multiplicity, and the limit exists and attains the floor if and only if the Riemann hypothesis holds for L(s,χ3)L(s,\chi_3) with all zeros simple. The constant is known: under GRH it is the variance of the limiting distribution of Rubinstein and Sarnak for (log⁡x/x)R(x)(\log x/\sqrt x)R(x) (four times smaller than in their normalization E(x;3,2)−E(x;3,1)E(x;3,2)-E(x;3,1)), and its closed form is that of Fiorilli and Martin (there under GRH); it is also the variance VχV_\chi of Part~37, on the 6k±16k\pm1 race, where the prime-square mechanism of the bias is identified; what is added here is the unconditional floor with its equality case and exponential-growth law, the ledger correction of the counting race, and the exact measurement. The constant κ3\kappa_3 is the exact analogue, for the prime race, of the B\'aez-Duarte constant 2+γE−log⁡4π2+\gamma_E-\log4\pi of the Nyman--Beurling criterion. Third, we measure: from all prime powers up to 10810^8, by exact piecewise integration, the cumulative mean energy is 0.12010.1201 at x=108x=10^8 and lies 44 to 12%12\% above κ3\kappa_3 for 104≤x≤10810^4\le x\le10^8. This is consistent with the theorem and does not test it: the floor is a statement about an upper limit, obtained from the dichotomy between GRH and its failure, and means over single decades fall below κ3\kappa_3 (0.1020.102 on [104,105][10^4,10^5], 0.1130.113 on [107,108][10^7,10^8]). The second-order term is (mean−κ3)⋅U≈0.11(\text{mean}-\kappa_3)\cdot U\approx0.11 (between 0.090.09 and 0.130.13), the race analogue of the second-order constant K=2.5124797…K=2.5124797\ldots of the prime energy. The proofs of the energy law transplant the C2C_2 energy law for ζ\zeta to L(s,χ3)L(s,\chi_3), where the absence of a pole simplifies every step.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 12 pages, 1 figures. Ancillary scripts in ancillary/ reproduce the computations. Part 38 of the book-series Cross-Energy of Dirichlet Partial Sums (47 parts, Parts 0-46); formerly the standalone note 'The energy of the Chebyshev race' (revised 1 Nov 2026). Renumbered 2 Nov 2026; mathematics unchanged. The paper states once, in its discussion, that it does not prove the Riemann hypothesis.

Provenance statement

The research programme and the choice of objects are the author's. The paper was written in November 2026 as the standalone note 'The energy of the Chebyshev race' and is Part 38 of the book-series Cross-Energy of Dirichlet Partial Sums; it was revised on 1 and 10 November 2026. It continues Part 37 (the 6k+-1 race) and transplants the energy law of Part 5 from zeta to L(s, chi_3); two estimates in the proof of Proposition 4.1 are imported from the zeta case without being rewritten. The constant kappa_3 is known (Rubinstein-Sarnak; Fiorilli-Martin). 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. All numbers in the paper are reproduced by race.py in the ancillary files (double precision, exact piecewise integration).

References

  1. L. Báez-Duarte, A strengthening of the Nyman--Beurling criterion for the Riemann hypothesis, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 14 (2003), 5--11.
  2. L. Báez-Duarte, M. Balazard, B. Landreau, E. Saias, Notes sur la fonction ζ\zeta de Riemann, 3, Adv. Math. 149 (2000), 130--144.
  3. C. Bays, R. H. Hudson, Details of the first region of integers xx with π3,2(x)<π3,1(x)\pi_{3,2}(x)<\pi_{3,1}(x), Math. Comp. 32 (1978), 571--576.
  4. P. L. Chebyshev, Lettre de M. le professeur Tchébychev à M. Fuss, sur un nouveau théorème relatif aux nombres premiers contenus dans les formes 4n+14n+1 et 4n+34n+3, Bull. Cl. Phys.-Math. Acad. Imp. Sci. St.-Pétersbourg 11 (1853), 208.
  5. H. Cramér, Ein Mittelwertsatz in der Primzahltheorie, Math. Z. 12 (1922), 147--153.
  6. H. Davenport, Multiplicative Number Theory , 3rd ed., Springer, 2000.
  7. D. Fiorilli, G. Martin, Inequities in the Shanks--Rényi prime number race: an asymptotic formula for the densities, J. reine angew. Math. 676 (2013), 121--212.
  8. A. Granville, G. Martin, Prime number races, Amer. Math. Monthly 113 (2006), 1--33.
  9. J. E. Littlewood, Sur la distribution des nombres premiers, C. R. Acad. Sci. Paris 158 (1914), 1869--1872.
  10. H. L. Montgomery, R. C. Vaughan, Hilbert's inequality, J. London Math. Soc. (2) 8 (1974), 73--82.
Show all 15 referencesShow fewer references
  1. M. Rubinstein, P. Sarnak, Chebyshev's bias, Experiment. Math. 3 (1994), 173--197.
  2. P. Caceres, Cross-Energy of Dirichlet Partial Sums. Part 5: The energy of the primes, preprint, 2026.
  3. P. Caceres, Cross-Energy of Dirichlet Partial Sums. Part 6: Second-order asymptotics and limiting fluctuations of the prime energy, preprint, 2026.
  4. P. Caceres, Cross-Energy of Dirichlet Partial Sums. Part 37: The 6k±16k\pm1 race: sieve structure, Chebyshev's bias and a GRH-equivalent inequality, preprint, 2026.
  5. P. Caceres, The cross-energy of Dirichlet partial sums and the zeros of the Riemann zeta function, consolidated paper, Hexagon preprint hexagon:2610.00161v2 (2026); Zenodo doi:10.5281/zenodo.23243641. (Its content is divided among Parts 1, 2, 5, 17 and 41 of the series.)hexagonDOI

Version history

  1. v1 · 12 pagesSubmitted by Pedro CaceresRevised 10 Nov 2026 after external examination (see CHANGES_v2.md). 12 pages, 1 figures. Ancillary scripts in ancillary/ reproduce the computations. Part 38 of the book-series Cross-Energy of Dirichlet Partial Sums (47 parts, Parts 0-46); formerly the standalone note 'The energy of the Chebyshev race' (revised 1 Nov 2026). Renumbered 2 Nov 2026; mathematics unchanged. The paper states once, in its discussion, that it does not prove the Riemann hypothesis.CurrentNov 11, 2026