On the distribution of consecutive square-free primitive roots modulo \(p\). (English) Zbl 1363.11090

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 \(p\).
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.


11N25 Distribution of integers with specified multiplicative constraints
11B50 Sequences (mod \(m\))
11L40 Estimates on character sums
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[1] D. R. Heath-Brown: The square sieve and consecutive square-free numbers. Math. Ann. 266 (1984), 251–259. · Zbl 0514.10038 · doi:10.1007/BF01475576
[2] H. Liu, W. Zhang: On the squarefree and squarefull numbers. J. Math. Kyoto Univ. 45 (2005), 247–255. · Zbl 1089.11052
[3] L. Mirsky: On the frequency of pairs of square-free numbers with a given difference. Bull. Amer. Math. Soc. 55 (1949), 936–939. · Zbl 0035.31301 · doi:10.1090/S0002-9904-1949-09313-8
[4] M. Munsch: Character sums over squarefree and squarefull numbers. Arch. Math. (Basel) 102 (2014), 555–563. · Zbl 1297.11097 · doi:10.1007/s00013-014-0658-9
[5] F. Pappalardi: A survey on k-freeness. Number Theory (S. D. Adhikari et al., eds.). Conf. Proc. Chennai, India, 2002, Ramanujan Mathematical Society, Ramanujan Math. Soc. Lect. Notes Ser. 1, Mysore, 2005, pp. 71–88.
[6] J. Rivat, A. Sárközy: Modular constructions of pseudorandom binary sequences with composite moduli. Period. Math. Hung. 51 (2005), 75–107. · Zbl 1111.11041 · doi:10.1007/s10998-005-0031-7
[7] H. N. Shapiro: Introduction to the Theory of Numbers. Pure and Applied Mathematics. Wiley-Interscience Publication, John Wiley & Sons. 12, New York, 1983. · Zbl 0515.10001
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