Euler Products at the Centre and Applications to Chebyshev's Bias

TL;DR

This paper investigates the behavior of partial Euler products of automorphic L-functions at the critical center, confirming a conjecture under GRH and Ramanujan assumptions, and applies results to Chebyshev's bias.

Contribution

It establishes an asymptotic for the partial Euler product at the center of the critical strip, confirming Kurokawa's conjecture under standard hypotheses.

Findings

Asymptotic behavior of partial Euler products at the critical point

Confirmation of Kurokawa's conjecture under GRH and Ramanujan assumptions

Results towards Chebyshev's bias in automorphic L-functions

Abstract

Let $\pi$ be an irreducible cuspidal automorphic representation of $\text{GL}_n(\mathbb A_\mathbb Q)$ with associated $L$-function $L(s, \pi)$. We study the behaviour of the partial Euler product of $L(s, \pi)$ at the center of the critical strip. Under the assumption of the Generalized Riemann Hypothesis for $L(s, \pi)$ and assuming the Ramanujan--Petersson conjecture when necessary, we establish an asymptotic, off a set of finite logarithmic measure, for the partial Euler product at the central point that confirms a conjecture of Kurokawa. As an application, we obtain results towards Chebyshev's bias in the recently proposed framework of Aoki-Koyama.