Chebyshev's bias and generalized Riemann hypothesis

Adel Alamadhi (1) , Michel Planat (2) , Patrick Solé (1)

1 Télécom ParisTech
2 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)

t is well known that li(x)>π(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)]>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 lmodk). The {\it Chebyshev's bias}(iii) is related to the generalized Riemann hypothesis (GRH) and occurs with a logarithmic density ≈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.

Domaines: Acoustique [physics.class-ph]; Matériaux; Micro et nanotechnologies/Microélectronique

Adel Alamadhi, Michel Planat, Patrick Solé. Chebyshev's bias and generalized Riemann hypothesis. Journal of Algebra, Number Theory: Advances and Applications, 2012, 8 (1 - 2), pp.41 - 55. ⟨hal-02300345⟩