| Title:
|
On the distribution of consecutive square-free primitive roots modulo $p$ (English) |
| Author:
|
Liu, Huaning |
| Author:
|
Dong, Hui |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
2 |
| Year:
|
2015 |
| Pages:
|
555-564 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A positive integer $n$ is called a square-free number if it is not divisible by a perfect square except $1$. Let $p$ be an odd prime. For $n$ with $(n,p)=1$, the smallest positive integer $f$ such that $n^f \equiv 1 \pmod p$ is called the exponent of $n$ modulo $p$. If the exponent of $n$ modulo $p$ is $p-1$, then $n$ is called a primitive root mod $p$. \endgraf Let $A(n)$ be the characteristic function of the square-free primitive roots modulo $p$. In this paper we study the distribution $$ \sum _{n\leq x}A(n)A(n+1), $$ and give an asymptotic formula by using properties of character sums. (English) |
| Keyword:
|
square-free |
| Keyword:
|
primitive root |
| Keyword:
|
square sieve |
| Keyword:
|
character sum |
| MSC:
|
11B50 |
| MSC:
|
11L40 |
| MSC:
|
11N25 |
| idZBL:
|
Zbl 06486965 |
| idMR:
|
MR3360445 |
| DOI:
|
10.1007/s10587-015-0194-1 |
| . |
| Date available:
|
2015-06-16T18:06:42Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144288 |
| . |
| Reference:
|
[1] Heath-Brown, D. R.: The square sieve and consecutive square-free numbers.Math. Ann. 266 (1984), 251-259. Zbl 0514.10038, MR 0730168, 10.1007/BF01475576 |
| Reference:
|
[2] Liu, H., Zhang, W.: On the squarefree and squarefull numbers.J. Math. Kyoto Univ. 45 (2005), 247-255. Zbl 1089.11052, MR 2161690, 10.1215/kjm/1250281988 |
| Reference:
|
[3] Mirsky, L.: On the frequency of pairs of square-free numbers with a given difference.Bull. Amer. Math. Soc. 55 (1949), 936-939. Zbl 0035.31301, MR 0031507, 10.1090/S0002-9904-1949-09313-8 |
| Reference:
|
[4] Munsch, M.: Character sums over squarefree and squarefull numbers.Arch. Math. (Basel) 102 (2014), 555-563. Zbl 1297.11097, MR 3227477, 10.1007/s00013-014-0658-9 |
| Reference:
|
[5] Pappalardi, F.: A survey on $k$-freeness.Number Theory S. D. Adhikari et al. Conf. Proc. Chennai, India, 2002 Ramanujan Mathematical Society, Ramanujan Math. Soc. Lect. Notes Ser. 1, Mysore (2005), 71-88. Zbl 1156.11338, MR 2131677 |
| Reference:
|
[6] Rivat, J., Sárközy, A.: Modular constructions of pseudorandom binary sequences with composite moduli.Period. Math. Hung. 51 (2005), 75-107. Zbl 1111.11041, MR 2194941, 10.1007/s10998-005-0031-7 |
| Reference:
|
[7] Shapiro, H. N.: Introduction to the Theory of Numbers.Pure and Applied Mathematics. Wiley-Interscience Publication John Wiley & Sons. 12, New York (1983). Zbl 0515.10001, MR 0693458 |
| . |
