A Positivity Conjecture Related to the Riemann Zeta Function

Abstract

According to two remarkable theorems of Nyman and Báez-Duarte, the Riemann hypothesis is equivalent to a simply stated criterion concerning least-squares approximation. In carrying out computations related to this criterion, we have observed a curious phenomenon: for no apparent reason, at least the first billion entries of a certain infinite triangular matrix associated with the Riemann zeta function are all positive. In this article, we describe the background leading to this observation, and make a conjecture.