@MISC{_countingprimes, author = {}, title = {COUNTING PRIMES IN RESIDUE CLASSES}, year = {} }

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Abstract

Abstract. We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method 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. 1.