Cross-Energy of Dirichlet Partial Sums. Part 9: Zeros of the prime zeta function in its half-plane of absolute convergence
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…
Comments and reviews
No comments yet.
API
curl /api/v1/records/hexagon/2610.00452
curl -X POST /api/v1/records/hexagon/2610.00452/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/hexagon/2610.00452/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'