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Cross-Energy of Dirichlet Partial Sums. Part 11: A two-sided criterion for the Riemann hypothesis from weighted prime sums, verified up to 2⋅10292\cdot10^{29}

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

Contributed by Pedro Caceres

Version 2 / Oct 31, 2026 / CC BY 4.0

Abstract

For N>1N>1 let C(N)=2∑n<NΛ(n)n−1/2log⁡(N/n)C(N)=2\sum_{n<N}\Lambda(n)n^{-1/2}\log(N/n) and let M(N)=R(N)−C(N)M(N)=R(N)-C(N), where R(N)=8N−5.3721834log⁡N−7.4013354+O(N−5/2)R(N)=8\sqrt N-5.3721834\log N-7.4013354+O(N^{-5/2}) is explicit. By a theorem of Suzuki, the Riemann hypothesis (RH) holds if and only if M(N)≥0M(N)\ge0 for all large NN, and then 0≤M(N)≤4(log⁡ξ)′′(12)=0.184840\le M(N)\le4(\log\xi)''(\frac12)=0.18484. We prove, with computer assistance, that this two-sided inequality holds for every real 1<N≤2.15⋅10291<N\le2.15\cdot10^{29}: up to 101010^{10} from the primes with correctly rounded arithmetic, and beyond from the first 45204520 zeros, certified with ball arithmetic, together with the verification of RH to 3⋅10123\cdot10^{12} and an explicit zero-density estimate. The same ingredients give, for every NN and without any hypothesis, −E(N)≤M(N)≤0.18484+E(N)-E(N)\le M(N)\le0.18484+E(N) with E(N)E(N) explicit and E(1020)<3⋅10−5E(10^{20})<3\cdot10^{-5}. We describe the local minima of MM exactly, through the slope α(x)=∑n≤xΛ(n)n−1/2−2x+c02\alpha(x)=\sum_{n\le x}\Lambda(n)n^{-1/2}-2\sqrt x+\frac{c_0}2. Under RH the margin is the variogram of α\alpha, and RH is equivalent to ∫1Xα2 dx/x=Xo(1)\int_1^X\alpha^2\,dx/x=X^{o(1)}. Without any hypothesis we prove an energy identity, which expresses the energy of the slope as a C-transform of the margin against the prime measure, and a lower bound for the mean energy that is attained exactly under RH with simple zeros. We then show what such arguments cannot do. Positive weights within qδ−1/2q^{\delta-1/2} of Λ(q)q−1/2\Lambda(q)q^{-1/2} satisfy the energy identity, the prime number theorem and Chebyshev's identity up to x1/2+δx^{1/2+\delta}, yet carry an off-line zero. So the structural inputs listed do not suffice, and a proof along these lines has to use Λ=μ∗log⁡\Lambda=\mu*\log exactly. Carrying that identity through the slope, α\alpha is numerically, to within 1%1\% in the range tested, a M\"obius-weighted sawtooth, and the exact argument reduces to a type~I sum of μ\mu against the fractional parts of x/dx/d, which is RH again.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 36 pages, 10 figures; ancillary scripts reproduce the computations; Part 11 of the 47-part book-series Cross-Energy of Dirichlet Partial Sums (Parts 0-46), block III (The margin and the constant B-hat); formerly a standalone note; a public version is the Hexagon preprint hexagon:2610.00661v1. Part 0 is the guide. The paper states the Riemann-hypothesis caveat once (2 Nov 2026: removed from the abstract and from a remark).

Provenance statement

Author’s original research under the program M=R-C; Claude (Anthropic) assisted with proofs, code, literature search and drafting; all computations are reproducible from the ancillary files and were verified by the author, who takes full responsibility.

Tools used

Anthropic
Claude OpusVersion 5.5
Python
PythonVersion 3.13

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  27. P. Caceres, Cross-Energy of Dirichlet Partial Sums. Part 29: Weil windows as density operators: mixtures, coherences and negativity, preprint, 2026.
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Version history

  1. v1 · 25 pagesSubmitted by Pedro CaceresInitial depositSupersededOct 29, 2026
  2. v2 · 36 pagesSubmitted by Pedro CaceresRevised 10 Nov 2026 after external examination (see CHANGES_v2.md). 36 pages, 10 figures; ancillary scripts reproduce the computations; Part 11 of the 47-part book-series Cross-Energy of Dirichlet Partial Sums (Parts 0-46), block III (The margin and the constant B-hat); formerly a standalone note; a public version is the Hexagon preprint hexagon:2610.00661v1. Part 0 is the guide. The paper states the Riemann-hypothesis caveat once (2 Nov 2026: removed from the abstract and from a remark).CurrentNov 11, 2026