Generalized Bombieri - Lagarias' theorem and generalized Li's criterion (1304.7895v3)

Abstract: We show that Li's criterion equivalent to the Riemann hypothesis, viz. the statement that the sums k_n=Sum_rho(1-(1-1/rho)^n) over Riemann xi-function zeroes as well as all derivatives lambda_n= (n-1)!^-1*d^n/dz^n(z^n-1*ln(xi(z))_(z=1)), where n=1, 2, 3..., should be non-negative if and only if the Riemann hypothesis holds true, can be generalized and the non-negativity 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=Sum_rho(1-(1-((rho-a)/(rho+a-1))^n) for any real a not equal to 1/2 as well as the derivatives (n-1)!^-1*d^n/dz^n((z-a)^n-1*ln(xi(z))_(z=(1-a))), n=1, 2, 3... for any a<1/2 are non-negative if and only if the Riemann hypothesis holds true (correspondingly, the same derivatives when a>1/2 should be non-positive for this). Similarly to Li's one, the theorem of Bombieri and Lagarias applied to certain multisets of complex numbers, is also generalized along the same lines.The results presented here are now available as a paper in Ukrainian Math. J., 64, 371-383, 2014.