Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 439-455, 2017
Density of solutions to quadratic congruences
Received December 31, 2015. First published May 5, 2017.
Abstract: 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 $n$.
Keywords: Dirichlet's theorem; asymptotic density; primes in arithmetic progression; squarefree number