On the distribution of consecutive square-free primitive roots modulo $p$

Liu, Huaning ; Dong, Hui

Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 555-564.

Résumé

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.

Détail

DOI : 10.1007/s10587-015-0194-1

Classification : 11B50, 11L40, 11N25
Keywords: square-free; primitive root; square sieve; character sum