# Une région explicite sans zéro pour les fonctions L de Dirichlet

Abstract : We establish the existence of explicit zero-free regions for the Riemann Zeta function. It never vanishes in the region on the left hand side of the axis $\Re s =1$~: \Re s \ge 1- \frac1(R_0 \log (|\Im s|+2)) with R_0=5.70175. The method is also successful in the more general case of the Dirichlet functions associated with a given modulus q. They never vanish in the region : \Re s \ge 1- \frac1(R_1 \log(q\max(1,|\Im s|))) with R_0=6.4355, except for at most one of them which should be real and which vanishes at most once in this part. Moreover, we precise that each function has at most four zeros in : \Re s \ge 1- \frac1(R_4 \log(q\max(1,|\Im s|))) where R_4=2.58208. The last part is dedicated to an application of these results to the distribution of prime numbers in an arithmetic progression (a+nq). The smallest of them (denoted by P(a,q)) satisfies~: P(a,q) \le \exp\big(\alpha(\log q)^2\big) where \alpha=6.95015 for q\ge10^6.
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https://tel.archives-ouvertes.fr/tel-00002695
Submitted on : Tuesday, April 8, 2003 - 10:25:21 AM
Last modification on : Thursday, November 12, 2020 - 3:27:20 AM
Long-term archiving on: : Tuesday, September 11, 2012 - 8:15:50 PM

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• HAL Id : tel-00002695, version 1

### Citation

Habiba Kadiri. Une région explicite sans zéro pour les fonctions L de Dirichlet. Mathématiques [math]. Université des Sciences et Technologie de Lille - Lille I, 2002. Français. ⟨tel-00002695⟩

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