## 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.

### MSC:

 11N25 Distribution of integers with specified multiplicative constraints 11B50 Sequences (mod $$m$$) 11L40 Estimates on character sums

### Keywords:

square-free; primitive root; square sieve; character sum
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### References:

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