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Cross-Energy of Dirichlet Partial Sums. Part 10: The Caceres Prime Function. A square-root damped prime weight, its exact Prime Zeta structure, and a dual spectrum of the Riemann zeros

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

\sloppy We study the \emph{Caceres Prime Function} \[ \C(s)=\sum_{n\ge 2}\frac{a(n)}{n^{s}},\qquad a(n)=\sum_{p^k\,\|\,n}\frac{\log p}{\sqrt{p^{k}}}, \] the Dirichlet series of the additive function that weights each prime factor pp of nn by log⁡p\log p divided by the square root of the exact power of pp dividing nn. \emph{Structure.} We prove that $\C=\zeta\K$, where the \emph{prime kernel}, $\K(s)=\sum_p\log p\,(1-p^{-s})/(p^{s+1/2}-1)$, is an exact telescoping series of shifted derivatives of the Prime Zeta function. So $\K$ is meromorphic in ℜs>−14\Re s>-\frac14, and the line ℜs=−14\Re s=-\frac14 is a natural boundary. Every zero 12+iγ\frac12+i\gamma of ζ\zeta produces two singular points of $\K$, at iγi\gamma and iγ/2i\gamma/2, which are poles unless 2γ2\gamma is again an ordinate with twice the multiplicity (respectively γ/2\gamma/2 an ordinate with half the multiplicity, for iγ/2i\gamma/2); the exponent 12\frac12 is the normalization that puts both families on the imaginary axis. More precisely, $\K(s)=-\frac{\zeta'}\zeta(s+\frac12)+\frac{\zeta'}\zeta(2s+\frac12)-\frac1{12s}+H(s)$, with HH holomorphic in a neighbourhood of ℜs≥0\Re s\ge0. \emph{RH criteria.} RH is equivalent to the holomorphy of $\K(s)-\frac1{s-1/2}+\frac1{2(s-1/4)}$ in ℜs>0\Re s>0. It is also equivalent to a bound O(xε)O(x^\varepsilon), or even a one-sided bound, for the remainder RbR_b of the kernel's partial sums. Both are exact reformulations of RH, not proofs. \emph{Mean value and explicit formula.} Unconditionally, $\sum_{n\le x}a(n)=\K(1)x+2\zeta(\frac12)\sqrt x+O(\sqrt x\,e^{-c_\ast\sqrt{\log x}})$, with the explicit bound 5.51x5.51\sqrt x for the error after the first term when x≥106x\ge10^6. Under RH, the Riesz mean of aa has an explicit formula in which every zero contributes the pure phases xiγx^{i\gamma} and xiγ/2x^{i\gamma/2} at constant amplitude. \emph{The Ces\`aro margin.} The kernel is Suzuki's one-variable margin MM seen at two scales. Unconditionally, Rb(x)=α(x)−α(x)+o(1)R_b(x)=\alpha(x)-\alpha(\sqrt x)+o(1), where α\alpha is the slope of the margin. Under RH, the kernel's Riesz margin is M(N)−2M(N)M(N)-2M(\sqrt N). Its fluctuation has energy 2(log⁡ξ)′′(12)2(\log\xi)''(\frac12), reduced by 2γ−22\gamma^{-2} for every pair of ordinates with γ′=2γ\gamma'=2\gamma; the near pair γ14≈2γ4\gamma_{14}\approx2\gamma_4 is visible in the data. \emph{Zeros of $\C$.} A computer-assisted proof shows that $\C$ has a zero near 1.379+8.639 i1.379+8.639\,i, in the half-plane of absolute convergence, and Bohr's theory gives ≫T\gg T such zeros up to height TT. So $\C$ is not an LL-function. The theorems are proved here, except that Section~\ref{sec:margin} uses the formula for the margin and the almost periodicity of the slope from Part~11. Uncertified computations are labelled as numerical observations.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 26 pages, 7 figures. Part 10 of a series of forty-seven on the cross-energy of Dirichlet partial sums (book edition of 2 November 2026, block II: Primes and Euler products; Part 0 is the guide, Part 1 the plan). Formerly a standalone note ("The Caceres Prime Function", October 2026); in the book it builds on Part 9 (the prime zeta function), Part 1 (the C-transform) and Part 11 (the Cesaro margin), and is used by Parts 11 and 37. All theorems are proved; uncertified computations are labelled as numerical observations. The ancillary archive (core.py, explicit.py, zeros.py, step*, rigor_meanvalue.py, rigor_rouche.py, ctransform.py, laplace.py, bridge_M*.py, variogram.py, slope_b.py) is the one of the standalone note, Caceres_Prime_Function_ancillary.zip.

Provenance statement

The functions studied here, the research programme and the questions are the author’s. Uncertified floating-point computations are labelled as numerical observations in the text; 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

References

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Show all 15 referencesShow fewer references
  1. P. Caceres, Cross-Energy of Dirichlet Partial Sums. Part 1: The C-transform and the cross-energy, preprint, 2026.
  2. P. Caceres, Cross-Energy of Dirichlet Partial Sums. Part 9: Zeros of the prime zeta function in its half-plane of absolute convergence, preprint, 2026; an earlier version is hexagon:2610.00452.hexagon
  3. P. Caceres, 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}, preprint, 2026; an earlier version is hexagon:2610.00661.hexagon
  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 · 26 pagesSubmitted by Pedro CaceresRevised 10 Nov 2026 after external examination (see CHANGES_v2.md). 26 pages, 7 figures. Part 10 of a series of forty-seven on the cross-energy of Dirichlet partial sums (book edition of 2 November 2026, block II: Primes and Euler products; Part 0 is the guide, Part 1 the plan). Formerly a standalone note ("The Caceres Prime Function", October 2026); in the book it builds on Part 9 (the prime zeta function), Part 1 (the C-transform) and Part 11 (the Cesaro margin), and is used by Parts 11 and 37. All theorems are proved; uncertified computations are labelled as numerical observations. The ancillary archive (core.py, explicit.py, zeros.py, step*, rigor_meanvalue.py, rigor_rouche.py, ctransform.py, laplace.py, bridge_M*.py, variogram.py, slope_b.py) is the one of the standalone note, Caceres_Prime_Function_ancillary.zip.CurrentNov 11, 2026