Density of solutions to quadratic congruences
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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 ≤ x with k prime factors such that a fixed quadratic equation has exactly 2k solutions modulo n.
KeywordsDirichlet’s theorem asymptotic density primes in arithmetic progression squarefree number
MSC 201011D45 11B25 11N37
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- H. L. Montgomery, R. C. Vaughan: Multiplicative Number Theory. I. Classical Theory. Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge, 2007.Google Scholar