Chebyshev's bias and generalized Riemann hypothesis

International audiencet is well known that li(x)&gtπ(x) (i) up to the (very large) Skewes' number x1∼1.40×10316 \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)≈x√logx[1+2∑γsin(γlogx)γ] (1). If Riemann hypothesis (RH) holds, (i) may be replaced by the equivalent statement li[ψ(x)]&gtπ(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)&gt0 (iii) holds for any x