Counting primes in residue classes

Abstract

We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko me- thod for computing π(x) can be used to compute 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.