Dirichlet's theorem; asymptotic density; primes in arithmetic progression; squarefree number
A classical result in number theory is Dirichlet's theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly $k$ prime factors for $k>1$. Building upon a proof by E. M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree $n\leq x$ with $k$ prime factors such that a fixed quadratic equation has exactly $2^k$ solutions modulo $n$.
 Hardy, G. H., Wright, E. M.: An Introduction to the Theory of Numbers
. Oxford University Press, Oxford (2008). MR 2445243
| Zbl 1159.11001
 Landau, E.: Sur quelques problèmes relatifs à la distribution des nombres premiers
. S. M. F. Bull. 28 (1900), 25-38 French. MR 1504359
| Zbl 31.0200.01