Counting Primes in Residue Classes

Marc Deléglise (1) , Pierre Dusart (2) , Xavier-François Roblot (3, 4)

1 ICJ - Institut Camille Jordan
2 LACO - Laboratoire d'Arithmétique, de Calcul formel et d'Optimisation
3 IGD - Institut Girard Desargues
4 CTN - Combinatoire, théorie des nombres

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

Domaines: Théorie des nombres [math.NT]

Marc Deléglise, Pierre Dusart, Xavier-François Roblot. Counting Primes in Residue Classes. Mathematics of Computation, 2004, 73 (247), pp.1565-1575. ⟨hal-00863138⟩