Chebyshev's bias and generalized Riemann hypothesis

Abstract

It is well known that $li(x)>π(x)$ (i) up to the (very large) Skewes' number $x_1 \sim 1.40 \times 10^{316}$ \cite{Bays00}. But, according to a Littlewood's theorem, there exist infinitely many $x$ that violate the inequality, due to the specific distribution of non-trivial zeros $γ$ of the Riemann zeta function $ζ(s)$, encoded by the equation $li(x)-π(x)\approx \frac{\sqrt{x}}{\log x}[1+2 \sum_γ\frac{\sin (γ\log x)}γ]$ (1). If Riemann hypothesis (RH) holds, (i) may be replaced by the equivalent statement $li[ψ(x)]>π(x)$ (ii) due to Robin \cite{Robin84}. A statement similar to (i) was found by Chebyshev that $π(x;4,3)-π(x;4,1)>0$ (iii) holds for any $x<26861$ \cite{Rubin94} (the notation $π(x;k,l)$ means the number of primes up to $x$ and congruent to $l\mod k$). The {\it Chebyshev's bias}(iii) is related to the generalized Riemann hypothesis (GRH) and occurs with a logarithmic density $\approx 0.9959$ \cite{Rubin94}. In this paper, we reformulate the Chebyshev's bias for a general modulus $q$ as the inequality $B(x;q,R)-B(x;q,N)>0$ (iv), where $B(x;k,l)=li[ϕ(k)*ψ(x;k,l)]-ϕ(k)*π(x;k,l)$ is a counting function introduced in Robin's paper \cite{Robin84} and $R$ resp. $N$) is a quadratic residue modulo $q$ (resp. a non-quadratic residue). We investigate numerically the case $q=4$ and a few prime moduli $p$. Then, we proove that (iv) is equivalent to GRH for the modulus $q$.