The cycle problem: an intriguing periodicity to the zeros of the Riemann zeta function

TL;DR

This paper uncovers a surprising periodicity in the sum of the real parts of the logarithmic integral evaluated at zeros of the Riemann zeta function, suggesting a new pattern in the distribution of these zeros.

Contribution

It introduces the cycle problem, highlighting an unexpected periodicity in sums involving the zeros of the Riemann zeta function, which may offer new insights into their distribution.

Findings

Identification of a periodic pattern in the sum of real parts of the logarithmic integral at zeta zeros

Potential implications for understanding the distribution of zeta zeros

New perspective on the structure of the zeros of the Riemann zeta function

Abstract

Summing the values of the real portion of the logarithmic integral of n^rho, where rho is one of a consecutive series of zeros of the Riemann zeta function, reveals an unexpected periodicity to the sum. This is the cycle problem.