{"id":"hexagon/2610.00161","pdf":"https://hexagonmath.org/pdf/2610.00161v2","url":"https://hexagonmath.org/abs/2610.00161v2","tags":["msc:11M26","msc:11M06","C2 Function","Dirichlet Sums","Riemann Zeta Function"],"score":0,"title":"THE CROSS-ENERGY OF DIRICHLET PARTIAL SUMS AND THE ZEROS OF THE RIEMANN ZETA FUNCTION","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Pedro Caceres"],"reviews":0,"subject":"math.NT","summary":"Let X_n(s) denote the sum of k^(-s) for k from 1 to n, and define the cross energy C_2(n,s) as |X_n(s)|^2 − |X_(n−1)(s)|^2 − n^(−2σ). At every point of the critical strip, the limit of C_2(n,s) is 1/(1/4 + t^2), zero, or positive infinity, according as σ = 1/2, σ > 1/2, or σ 0, which is equivalent to RH. An off-line pair reduces S precisely inside a disc; at abscissa σ, the failure set up to height T has measure at most N(σ,T + 1). Moreover, S is Weil’s functional evaluated at a Poisson kernel, and its truncation admits an exact finite expression involving primes up to e^L. We test each…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Sonnet","ChatGPT","Python"],"discussed":0,"published":"2026-10-09","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}