Chebyshev's Bias for Ramanujan's $\tau$-function via the Deep Riemann Hypothesis

TL;DR

This paper demonstrates, under the assumption of the Deep Riemann Hypothesis, that Ramanujan's tau-function exhibits a bias towards positive values, analogous to Chebyshev's bias.

Contribution

It establishes a novel connection between the Deep Riemann Hypothesis and the sign bias of Ramanujan's tau-function.

Findings

Weighted sums of tau-function tend to be positive

The bias is analogous to Chebyshev's bias

Results depend on the Deep Riemann Hypothesis

Abstract

The authors assume the Deep Riemann Hypothesis to prove that a weighted sum of Ramanujan's $\tau$-function has a bias to being positive. This phenomenon is an analogue of Chebyshev's bias.