A sharpening of Li's criterion for the Riemann Hypothesis

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, ${\lambda }_n\sim n(A\log n+B)$ for $n\to \infty$ (with explicit $A>0$ and $B$). The approach also holds for more general zeta or $L$-functions.

1 Latex file, 5 pages, submitted to C.R. Acad. Sci. (Paris) Sér. I. V2: notation corrected in eq.(7); eq.(9) made more precise