Generalized Bombieri–Lagarias’ Theorem and Generalized Li’s Criterion with its Arithmetic Interpretation

Abstract

We show that Li’s criterion equivalent to the Riemann hypothesis, i.e., the statement that the sums $$ {k}_n={\Sigma}_{\rho}\left(1-{\left(1-\frac{1}{\rho}\right)}^n\right) $$ over zeros of the Riemann xi-function and the derivatives $$ \begin{array}{ccc}\hfill {\lambda}_n\equiv \frac{1}{\left(n-1\right)!}\frac{d^n}{d{z}^n}{\left.\left({z}^{n-1} \ln \left(\xi (z)\right)\right)\right|}_{z=1},\hfill & \hfill \mathrm{where}\hfill & \hfill n=1,2,3,\dots, \hfill \end{array} $$ are nonnegative if and only if the Riemann hypothesis is true, can be generalized and the nonnegativity of certain derivatives of the Riemann xi-function estimated at an arbitrary real point a, except a = 1/2, can be used as a criterion equivalent to the Riemann hypothesis. Namely, we demonstrate that the sums $$ {k}_{n,a}={\Sigma}_{\rho}\left(1-{\left(\frac{\rho -a}{\rho +a-1}\right)}^n\right) $$ for any real a such that a < 1/2 are nonnegative if and only if the Riemann hypothesis is true (correspondingly, the same derivatives with a > 1/2 should be nonpositive). The arithmetic interpretation of the generalized Li’s criterion is given. Similarly to Li’s criterion, the theorem of Bombieri and Lagarias applied to certain multisets of complex numbers is also generalized along the same lines.