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Keywords:
square-free; primitive root; square sieve; character sum
square-free; primitive root; square sieve; character sum
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.
References:
[1] Heath-Brown, D. R.: The square sieve and consecutive square-free numbers. Math. Ann. 266 (1984), 251-259. DOI 10.1007/BF01475576 | MR 0730168 | Zbl 0514.10038
[2] Liu, H., Zhang, W.: On the squarefree and squarefull numbers. J. Math. Kyoto Univ. 45 (2005), 247-255. DOI 10.1215/kjm/1250281988 | MR 2161690 | Zbl 1089.11052
[3] Mirsky, L.: On the frequency of pairs of square-free numbers with a given difference. Bull. Amer. Math. Soc. 55 (1949), 936-939. DOI 10.1090/S0002-9904-1949-09313-8 | MR 0031507 | Zbl 0035.31301
[4] Munsch, M.: Character sums over squarefree and squarefull numbers. Arch. Math. (Basel) 102 (2014), 555-563. DOI 10.1007/s00013-014-0658-9 | MR 3227477 | Zbl 1297.11097
[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. MR 2131677 | Zbl 1156.11338
[6] Rivat, J., Sárközy, A.: Modular constructions of pseudorandom binary sequences with composite moduli. Period. Math. Hung. 51 (2005), 75-107. DOI 10.1007/s10998-005-0031-7 | MR 2194941 | Zbl 1111.11041
[7] Shapiro, H. N.: Introduction to the Theory of Numbers. Pure and Applied Mathematics. Wiley-Interscience Publication John Wiley & Sons. 12, New York (1983). MR 0693458 | Zbl 0515.10001
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