THE CROSS-ENERGY OF DIRICHLET PARTIAL SUMS AND THE ZEROS OF THE RIEMANN ZETA FUNCTION
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…
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