Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 439-455, 2017


Density of solutions to quadratic congruences

Neha Prabhu

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
Classification MSC:  11D45, 11B25, 11N37


References:
[1] G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (2008). MR 2445243 | Zbl 1159.11001
[2] H. Kornblum, E. Landau: Über die Primfunktionen in einer arithmetischen Progression. Math. Zeitschr. 5 (1919), 100-111. (In German). DOI 10.1007/BF01203156 | MR 1544375 | Zbl 47.0154.02
[3] E. Landau: Sur quelques problèmes relatifs à la distribution des nombres premiers. S. M. F. Bull. 28 (1900), 25-38. (In French). MR 1504359 | Zbl 31.0200.01
[4] H. L. Montgomery, R. C. Vaughan: Multiplicative Number Theory. I. Classical Theory. Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge (2007). DOI 10.1017/CBO9780511618314 | MR 2378655 | Zbl 1142.11001
[5] C. Pomerance: On the distribution of amicable numbers. J. Reine Angew. Math. 293/294 (1977), 217-222. DOI 10.1515/crll.1977.293-294.217 | MR 0447087 | Zbl 0349.10004
[6] P. Ribenboim: The New Book of Prime Number Records. Springer, New York (1996). DOI 10.1007/978-1-4612-0759-7 | MR 1377060 | Zbl 0856.11001
[7] E. M. Wright: A simple proof of a theorem of Landau. Proc. Edinb. Math. Soc., II. Ser. 9 (1954), 87-90. DOI 10.1017/S0013091500021349 | MR 0065579 | Zbl 0057.28601

Affiliations:   Neha Prabhu, Indian Institute of Science Education and Research, Dr Homi Bhabha Rd, NCL Colony, Pashan, Pune, Maharashtra 411008, India, e-mail: neha.prabhu@students.iiserpune.ac.in


 
PDF available at: