{"id":"hexagon/2610.00452","pdf":"https://hexagonmath.org/pdf/2610.00452v1","url":"https://hexagonmath.org/abs/2610.00452v1","tags":["math.CV","msc:11M41","C2 Function","Dirichlet Sums","Riemann Zeta Function"],"score":0,"title":"Cross-Energy of Dirichlet Partial Sums. Part 9: Zeros of the prime zeta function in its half-plane of absolute convergence","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Pedro Caceres"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","Python"],"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}