A sharpening of Li's criterion for the Riemann Hypothesis

en · official · sha256:a0b797c9b7fede4bc04358cce400728d89bf20c063f7107888c2afff3a34d142

Abstract. Exact and asymptotic formulae are displayed for the coefficients lambda_n used in Li's criterion for the Riemann Hypothesis. In particular, we argue that if (and only if) the Hypothesis is true, lambda_n ~ n(A log n + B) for n -> infinity (with explicit A > 0 and B). The approach also holds for more general zeta or L-functions.

Referencing norms: li-asserted-via-voros

en · official · sha256:58e7cc13538d66893cf97e803fab5fec75ad763de281df0c337c1b66c69b09f0

Alternative expressions are presented for the coefficients lambda_n introduced by Li [7, 1] to recast the Riemann Hypothesis as lambda_n > 0 (for all n): a first representation (6) as a finite oscillatory sum, then a closely related integral representation (7), upon which the saddle-point method finally yields definite n -> +infinity asymptotic estimates in one of two sharply distinct forms, (9) or (10), depending on the falsity or truth of the Riemann Hypothesis.

en · official · sha256:25e32f90bc404bdc89c2ef3ffbe32c8afa6153f7680ffe25f4b130b7f46f5059

The coefficients lambda_n are defined as lambda_n = sum_rho [1 - (1 - 1/rho)^n], or equivalently via the generating function d/dz log Xi(1/(1-z)) = sum_{n=1}^infinity lambda_n z^(n-1), (Xi(s) = s(s-1) Gamma(s/2) pi^(-s/2) zeta(s)).

en · official · sha256:8cbb588cce421ef8d7d3c75164a3e25a18e3be28a7b44cecfe56f287f81296fe

A sharpening of Li's criterion for the Riemann Hypothesis Andre Voros

Referencing norms: LiSource, VorosSource