Nonlocal correlation in L-functions

Abstract

We identify a nonlocal correlation structure in L-functions. This structure involves very long and infinite-range correlations between values of logarithmic L-functions, where the correlation strongly depends upon the presence of a multiplicative relationship between the two points in question on the complex plane. This correlation sharply jumps if the two points share a multiplicative relationship and takes much lower values otherwise, demonstrating a previously undescribed type of long-range order in L-functions. We leverage this correlation structure to provide a novel set of identities that relate the distribution of M\"obius function solutions, sums over the non-divisors of the integers, and the polylogarithms.