Chebyshev's bias and generalized Riemann hypothesis

Abstract : It is well known that $\mbox{li}(x)>\pi(x)$ (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 $\gamma$ of the Riemann zeta function $\zeta(s)$, encoded by the equation $\mbox{li}(x)-\pi(x)\approx \frac{\sqrt{x}}{\log x}[1+2 \sum_{\gamma}\frac{\sin (\gamma \log x)}{\gamma}]$ (1). If Riemann hypothesis (RH) holds, (i) may be replaced by the equivalent statement $\mbox{li}[\psi(x)]>\pi(x)$ (ii) due to Robin \cite{Robin84}. A statement similar to (i) was found by Chebyshev that $\pi(x;4,3)-\pi(x;4,1)>0$ (iii) holds for any $x<26861$ \cite{Rubin94} (the notation $\pi(x;k,l)$ 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 $B(x;k,l)=\mbox{li}[\phi(k)*\psi(x;k,l)]-\phi(k)*\pi(x;k,l)$ 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 $q$.
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  • HAL Id : hal-00650320, version 1
  • ARXIV : 1112.2398

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Adel Alahmadi, Michel Planat, Patrick Solé. Chebyshev's bias and generalized Riemann hypothesis. Journal of Algebra, Number Theory: Advances and Applications, 2013, 8 (1-2), pp.41-55. ⟨hal-00650320⟩

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