Work

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

Year: 2025

Type: article

Abstract: Let $\pi$ be an irreducible cuspidal automorphic representation of ${\mathrm{GL}}_n(\mathbb{A}_{\mathbb{Q}})$ with associated L-function $L(s, \pi)$ . We study the behaviour of the partial Euler product of $L(s, \pi)$ at the centre of the critical strip. Under the assumption of the Generalised Riemann Hypothesis for $L(s, \pi)$ in conjunction with the Ramanujan–Petersson conjecture as necessary, we establish an asymptotic, off a set of finite logarithmic measure, for the partial Euler product at the central point, which confirms a conjecture of Kurokawa (2012). As an application, we obtain results towards Chebyshev’s bias in the recently proposed framework of Aoki–Koyama (2023).

Source: Mathematical Proceedings of the Cambridge Philosophical Society

DOI: 10.1017/s0305004125000234

Author: ARSHAY SHETH

Institution: University of Warwick

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

Funder: European Commission