| Title:
|
Density of solutions to quadratic congruences (English) |
| Author:
|
Prabhu, Neha |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
439-455 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $n$. (English) |
| Keyword:
|
Dirichlet's theorem |
| Keyword:
|
asymptotic density |
| Keyword:
|
primes in arithmetic progression |
| Keyword:
|
squarefree number |
| MSC:
|
11B25 |
| MSC:
|
11D45 |
| MSC:
|
11N37 |
| idZBL:
|
Zbl 06738530 |
| idMR:
|
MR3661052 |
| DOI:
|
10.21136/CMJ.0.0712-15 |
| . |
| Date available:
|
2017-06-01T14:29:41Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146767 |
| . |
| Reference:
|
[1] Hardy, G. H., Wright, E. M.: An Introduction to the Theory of Numbers.Oxford University Press, Oxford (2008). Zbl 1159.11001, MR 2445243 |
| Reference:
|
[2] Kornblum, H., Landau, E.: Über die Primfunktionen in einer arithmetischen Progression.Math. Zeitschr. 5 (1919), 100-111 German. Zbl 47.0154.02, MR 1544375, 10.1007/BF01203156 |
| Reference:
|
[3] Landau, E.: Sur quelques problèmes relatifs à la distribution des nombres premiers.S. M. F. Bull. 28 (1900), 25-38 French. Zbl 31.0200.01, MR 1504359 |
| Reference:
|
[4] Montgomery, H. L., Vaughan, R. C.: Multiplicative Number Theory. I. Classical Theory.Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge (2007). Zbl 1142.11001, MR 2378655, 10.1017/CBO9780511618314 |
| Reference:
|
[5] Pomerance, C.: On the distribution of amicable numbers.J. Reine Angew. Math. 293/294 (1977), 217-222. Zbl 0349.10004, MR 0447087, 10.1515/crll.1977.293-294.217 |
| Reference:
|
[6] Ribenboim, P.: The New Book of Prime Number Records.Springer, New York (1996). Zbl 0856.11001, MR 1377060, 10.1007/978-1-4612-0759-7 |
| Reference:
|
[7] Wright, E. M.: A simple proof of a theorem of Landau.Proc. Edinb. Math. Soc., II. Ser. 9 (1954), 87-90. Zbl 0057.28601, MR 0065579, 10.1017/S0013091500021349 |
| . |
