Analysis of Voros criterion: what derivatives involving the logarithm of the Riemann xi-function at z=1/2 should be non-negative for the Riemann hypothesis holds true

TL;DR

This paper investigates sums involving derivatives of the logarithm of the Riemann xi-function at 1/2, establishing their non-negativity as equivalent to the Riemann hypothesis, and expresses these sums through non-negative even powers of the zeros.

Contribution

It provides a new formulation linking derivatives of the xi-function logarithm to the Riemann hypothesis using sums with non-negative even powers of zeros.

Findings

Derives expressions involving only non-negative even powers of zeros

Shows equivalence between non-negativity of derivatives and Riemann hypothesis

Provides explicit formulas with binomial coefficient-based coefficients

Abstract

Recently, Voros has found the sums involving certain powers of z-1/2, which, when taken over Riemann xi-function zeroes /rho, must be positive for the Riemann hypothesis holds true and vice versa. Here we analyze these sums, write them as expressions involving only non-negative even powers of /rho-1/2, and show that the Riemann hypothesis is equivalent for the non-negativity of the derivatives (1/(2n-1)!)*d^(2n)/dz^(2n)(F_(2n)(z)*ln(/xi(z))) at z=1/2 where F_(2n)=4*Sum_(k=0)^(n-1)((n-k)*A_(k,n)*(z-1/2)^(2k) with the coefficients A_(k,n)=a^(2k-2n)*Sum_(l=k)^(n)(C_(2n)^(2l)*C_(l)^(k)), C_(j)^(m) are binomial coefficients, for any n=1, 2, 3... and any real a>1/14.