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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 1 / Oct 22, 2026 / CC BY 4.0

Abstract

The prime zeta function P(s) = ∑ₚ p⁻ˢ has zeros in its half-plane of absolute convergence. Belovas, Čepaitytė and Sabaliauskas proved that P(s) ≠ 0 for Re(s) > σₚ = 1.7795446535…, where σₚ solves P(σ) = 2¹⁻σ. We show that this bound is sharp: the closure of the real parts of the zeros in Re(s) > 1 is exactly [1, σₚ]. There are no zeros on Re(s) = σₚ, while zeros occur arbitrarily close to every line Re(s) = σ with 1 < σ < σₚ (Bohr’s method). By Jessen’s theorem, the density of real parts is J″(σ)/(2π), where J(σ) = −σ log 2 + 𝔼[log⁺(2^σ |Yσ|)], Yσ = ∑ₚ≥₃ p⁻σ eⁱθₚ, and the phases θₚ are independent and uniformly distributed. The function J is smooth, σₚ = inf{σ : J(σ) + σ log 2 = 0}, and the density vanishes at σₚ faster than any power. The argument principle, implemented with ball arithmetic, certifies exactly 68 zeros with 1.1 ≤ Re(s) ≤ 1.85 below height 2000 (70.0 predicted), and 12 zeros with Re(s) ≥ 1.5 below height 5000 (12.0 predicted). A heuristic large-deviation extrapolation predicts zeros with real part above 1.7 approximately once every 6 × 10⁶ units of height. In ½ < Re(s) < 1, P is single-valued on the half-plane with (½, 1] removed if and only if the Riemann hypothesis holds (a reformulation of a classical equivalence). Its zeros are canonical below height 3 × 10¹² and solve ζ(s) = exp(A(s)), A(s) = ∑ₘ≥₂ P(ms)/m, with |P(s)| < 2π. Below height 600, their counts for Re(s) ≥ 0.7 agree numerically with the random model. The zeros arise from cancellations between 2⁻ˢ and the contributions of the remaining primes, the prime-sum analogue of Euler-product interference behind the off-line zeros of the Davenport–Heilbronn function. We do not prove the Riemann hypothesis.

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out of the author’s 2018 numerical study of the prime zeta function (PrimeZeta, DNA Primes Math Blog, 2018; and the unpublished note Riemann zeta function – nine propositions, 2018), which computed and plotted zeros of P(s) and reported clusters both to the right and to the left of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

  1. I. Belovas, R. Čepaitytė, M. Sabaliauskas, On the zero-free region and the distribution of zeros of the prime zeta function, An. Şt. Univ. Ovidius Constanţa Ser. Mat. 33 (2025), no. 2, 27--44, doi:10.2478/auom-2025-0017.DOI
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  6. P. Caceres, Cross-energy of Dirichlet partial sums. Part 1: The C-transform and the cross-energy, preprint, 2026.

Version history

  1. v1Submitted by Pedro CaceresInitial depositCurrentOct 21, 2026