# Chebyshev's bias for analytic L-functions

## TL;DR

This paper develops a broad framework for analyzing Chebyshev's bias in prime number races related to general L-functions, extending previous results and weakening key hypotheses like GRH.

## Contribution

It introduces a generalized approach to Chebyshev's bias for a wide class of L-functions, relaxing traditional assumptions and establishing new existence results for logarithmic densities.

## Key findings

- Established the existence of logarithmic density for prime sums related to general L-functions.
- Extended Chebyshev's bias analysis to new classes of L-functions beyond previous cases.
- Weakened hypotheses such as GRH needed for bias existence results.

## Abstract

In this paper we discuss the generalizations of the concept of Chebyshev's bias from two perspectives. First we give a general framework for the study of prime number races and Chebyshev's bias attached to general $L$-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet $L$-functions and Hasse--Weil $L$-functions of elliptic curves over $\mathbf{Q}$. This also apply to new Chebyshev's bias phenomena that were beyond the reach of the previously known cases. In addition we weaken the required hypotheses such as GRH or linear independence properties of zeros of $L$-functions. In particular we establish the existence of the logarithmic density of the set $\lbrace x\geq 2 : \sum_{p\leq x} \lambda_{f}(p) \geq 0 \rbrace$ for coefficients $(\lambda_{f}(p))$ of general $L$-functions conditionally on a much weaker hypothesis than was previously known.