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Cross-Energy of Dirichlet Partial Sums. Part 9: Zeros of the prime zeta function in its half-plane of absolute convergence

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 22, 2026 / CC BY 4.0

Abstract

The prime zeta function P(s)=∑pp−sP(s)=\sum_pp^{-s} has zeros in its half-plane of absolute convergence. Belovas, \v{C}epaityt\.{e} and Sabaliauskas proved that P≠0P\ne0 for Re⁡s>σP=1.7795446535…\operatorname{Re}s>\sigma_P=1.7795446535\ldots, the solution of P(σ)=21−σP(\sigma)=2^{1-\sigma}. We show that this is sharp: the closure of the real parts of the zeros in Re⁡s>1\operatorname{Re}s>1 is exactly [1,σP][1,\sigma_P], there are no zeros on Re⁡s=σP\operatorname{Re}s=\sigma_P, and zeros occur arbitrarily close to every line Re⁡s=σ\operatorname{Re}s=\sigma with 1<σ<σP1<\sigma<\sigma_P (Bohr's method). By Jessen's theorem the density of real parts is J′′(σ)/2πJ''(\sigma)/2\pi, where J(σ)=−σlog⁡2+Elog⁡+(2σ∣Yσ∣)J(\sigma)=-\sigma\log2+\mathbb E\log^+(2^\sigma|Y_\sigma|) and Yσ=∑p≥3p−σeiθpY_\sigma=\sum_{p\ge3}p^{-\sigma}e^{i\theta_p} has independent uniform phases. JJ is smooth, σP=inf⁡{σ:J(σ)+σlog⁡2=0}\sigma_P=\inf\{\sigma:J(\sigma)+\sigma\log2=0\}, and the density vanishes at σP\sigma_P faster than any power. The argument principle in ball arithmetic certifies exactly 6868 zeros with 1.1≤Re⁡s≤1.851.1\le\operatorname{Re}s\le1.85 below height 20002000 (70.070.0 predicted) and 1212 with Re⁡s≥1.5\operatorname{Re}s\ge1.5 below height 50005000 (12.012.0 predicted). A heuristic large-deviation extrapolation predicts zeros with real part above 1.71.7 about once every 6⋅1066\cdot10^6 units of height. In 12<Re⁡s<1\frac12<\operatorname{Re}s<1, PP is single-valued on the half-plane minus (12,1](\frac12,1] if and only if RH holds (a reformulation of a classical equivalence), its zeros are canonical below height 3⋅10123\cdot10^{12}, and they are the solutions of ζ(s)=eA(s)\zeta(s)=e^{A(s)}, A(s)=∑m≥2P(ms)/mA(s)=\sum_{m\ge2}P(ms)/m, with ∣P(s)∣<2π|P(s)|<2\pi. Below height 600600 their counts for Re⁡s≥0.7\operatorname{Re}s\ge0.7 agree numerically with the random model. The zeros are cancellations between 2−s2^{-s} and the remaining primes, the prime-sum analogue of the Euler-product interference behind the off-line zeros of the Davenport--Heilbronn function. Version~2 adds the critical line: the partial prime zeta function weights each zero by 1/((ρ−s)log⁡x)1/((\rho-s)\log x), and a Gaussian slice of the primes weights it by e−ε2(γ−T)2e^{-\varepsilon^2(\gamma-T)^2}. In a numerical experiment at height 10001000, the partial prime zeta function aimed at an off-line pair detects it sooner than a wide slice (ε=0.02\varepsilon=0.02) and later than a narrow one (ε=1\varepsilon=1); aimed 3030 away, the wide slice is the best of the three.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). Version 2 (2 Nov 2026). 17 pages, 6 figures, 8 tables; ancillary Python scripts reproduce the tables and figures. Part 9 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). We do not prove the Riemann hypothesis. 2 Nov 2026: the paper states the RH caveat once, in the Discussion (Section 7 in version 2); the abstract and introduction no longer repeat it (no other change).

Provenance statement

Comments: 11 pages, 3 figures, 7 tables; ancillary Python scripts reproduce the tables and figures. Part 9 of a series of seventeen on the cross-energy of Dirichlet partial sums. We do not prove the Riemann hypothesis. This paper grew out of the author’s 2018 numerical study of the prime zeta function (PrimeZeta, DNA Primes Math Blog, 2018; and the unpublished note Riemann zeta function – nine propositions, 2018), which computed and plotted zeros of P(s) and reported clusters both to the right and to the left of Re(s) = 1. The present version revisits that computation rigorously. It shows that the zeros with 1 < Re(s) < 1.6 found in 2018 are genuine, while those with real part below 1 depend on the branch of log ζ used by the software. It also proves that the zero-free bound σₚ of Belovas, Čepaitytė and Sabaliauskas is sharp, and certifies the counts using ball arithmetic.

Tools used

Anthropic
Claude OpusVersion 5.5
Python
PythonVersion 3.13

References

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  6. 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.
  7. P. Caceres, Cross-Energy of Dirichlet Partial Sums. Part 1: The C-transform and the cross-energy, preprint, 2026.
  8. P. Caceres, Cross-Energy of Dirichlet Partial Sums. Part 8: The Euler product inside the critical strip, preprint, 2026.
  9. P. Caceres, Cross-Energy of Dirichlet Partial Sums. Part 46: Controls for the Riemann hypothesis: Davenport--Heilbronn, Beurling systems and function fields, preprint, 2026.

Version history

  1. v1 · 11 pagesSubmitted by Pedro CaceresInitial depositSupersededOct 21, 2026
  2. v2 · 17 pagesSubmitted by Pedro CaceresRevised 10 Nov 2026 after external examination (see CHANGES_v2.md). Version 2 (2 Nov 2026). 17 pages, 6 figures, 8 tables; ancillary Python scripts reproduce the tables and figures. Part 9 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). We do not prove the Riemann hypothesis. 2 Nov 2026: the paper states the RH caveat once, in the Discussion (Section 7 in version 2); the abstract and introduction no longer repeat it (no other change).CurrentNov 11, 2026