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Counting primes in residue classes

Authors:	Marc Delé glise Pierre Dusart Xavier-Franç ois Roblot

Institution:	Institut Girard Desargues, Université Lyon I, 21, avenue Claude Bernard, 69622 Villeurbanne Cedex, France ; LACO, Département de Mathématiques, avenue Albert Thomas, 87060 Limoges Cedex, France ; Institut Girard Desargues, Université Lyon I, 21, avenue Claude Bernard, 69622 Villeurbanne Cedex, France

Abstract:	We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing $\pi(x)$ can be used for computing efficiently $\pi(x,k,l)$ , the number of primes congruent to modulo up to . As an application, we computed the number of prime numbers of the form $4n \pm 1$ less than for several values of up to $10^{20}$ and found a new region where $\pi(x,4,3)$ is less than $\pi(x,4,1)$ near $x = 10^{18}$ .

Keywords:	Prime numbers residue classes Chebyshev's bias

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