A positivity conjecture related to the Riemann zeta function

TL;DR

This paper explores a numerical phenomenon related to the Riemann zeta function, observing positivity in a large matrix and proposing a conjecture connected to the Riemann hypothesis.

Contribution

It introduces a new positivity conjecture based on computational observations linked to the Riemann hypothesis and related approximation criteria.

Findings

First billion entries of the matrix are positive

Observation suggests a potential underlying positivity property

Conjecture may provide new insights into the Riemann hypothesis

Abstract

According to two remarkable theorems of Nyman and B\'aez-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 to the Riemann zeta function are all positive. In this article we describe the background leading to this observation, and make a conjecture.