On the distribution of square-full and cube-full primitive roots
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On the distribution of square-full and cube-full primitive roots Teerapat Srichan1
© Akadémiai Kiadó, Budapest, Hungary 2020
Abstract A positive integer n is called an r -full integer if for all primes p | n we have pr | n. Let p be an odd prime. For gcd(n, p) = 1, the smallest positive integer f such that n f ≡ 1 (mod p) is called the exponent of n modulo p. If f = p − 1 then n is called a primitive root modulo p. Let Tr (n) be the characteristic function of the r -full primitive roots modulo p. In this paper we derive the asymptotic formula for the following sums T2 (n), T3 (n), n≤x
n≤x
by using properties of character sums. Keywords Character sums · Cube-full integers · Primitive roots · Square-full integers Mathematics Subject Classification 11N25 · 11B50
1 Introduction and results The problem of counting the primitive roots that are square-full is a topic in analytic number theory. In 1983 Shapiro [3] investigated square-full primitive roots and showed that
T2 (n) =
n≤x
φ( p − 1) √ c x + O(x 1/3 p 1/6 (log p)1/3 2ω( p−1) ) , p−1
(1.1)
where φ(n) is Euler’s function, ω(n) denotes the number of distinct prime divisors of n, and μ2 (q) 1 . c =2 1− p q 3/2 (q| p)=−1
Very recently, Munsch and Trudgian [5] improved on the error term in (1.1) and showed that
B 1
Teerapat Srichan [email protected] Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok, Thailand
123
T. Srichan
n≤x
−
φ( p − 1) 1 1 1 −1 C p x 1/2 1+ + 2 p−1 ζ (3) p p + O(x 1/3 (log x) p 1/9 (log p)1/6 2ω( p−1) ) ,
T2 (n) =
(1.2)
1 √
where C p p 8 e . Munsch and Trudgian used the character estimate of Burgess (see [1]) and Lemma 1.3 of [4] to prove (1.2). They only consider the contribution of the principal and the quadratic characters. It would be interesting to see whether their method could be improved to consider the cubic characters. In this paper we shall improve the result in (1.2) by using character sums. The considered characters are the principal, quadratic and cubic characters. The essential lemmas follow from the proof in Theorem 2.1 in [6]. We obtain the following theorem. Theorem 1.1 For a given odd prime p ≤ x 1/5 , φ( p − 1) L(3/2, χ0 ) − L(3/2, χ1 ) T2 (n) = x 1/2 p L(3, χ0 ) n≤x φ( p − 1) L(2/3, χ0 ) − L(2/3, χ22 ) + x 1/3 p L(2, χ0 ) + O x 1/6 φ( p − 1)3ω1,3 ( p−1) p 1/2+ , here χ0 , χ1 = χ0 , and χ2 = χ0 denote respectively the principal, quadratic and cubic character modulo p. The terms with the cubic characters only occur if 3| p − 1. Finally, ω1,3 (n) denotes the number of distinct prime q ≡ 1 (mod 3) and q are divisors of n. Remark 1.2 The result in Theorem 1.1 improves on (1.2) when p < x 3/25 . Remark 1.3 Munsch and Trudgian’s result shows that for all sufficiently large p there is a positive square-full primitive root less than p. Cohen and Trudgian [2] conjectured that this may in fact hold for p > 1052041. It would be interesting to see whether the result in Theorem 1.1 could prove this conjecture. It is natural to try study cube-full pr
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