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Let k be a fixed integer. We study the asymptotic formula of R(H,r,k), which is the number of positive integer solutions 1 ⩽ x, y, z ⩽ H such that the polynomial x2 + y2 + z2 + k is r-free. We obtained the asymptotic formula of R(H, r, k) for all r ⩾ 2. Our result is new even in the case r = 2. We proved that R(H, 2, k) = ckH3 + O(H9/4+ε), where ck > 0 is a constant depending on k. This improves upon the error term O(H7/3+ε) obtained by G.-L. Zhou, Y. Ding (2022). [ABSTRACT FROM AUTHOR]

We show that for any given integer k there exist infinitely many consecutive square-free numbers of the type x2 + y2 + z2 + k, x2 + y2 + z2 + k + 1. We also establish an asymptotic formula for 1 ⩽ x, y, z ⩽ H such that x2 + y2 + z2 + k, x2 + y2 + z2 + k + 1 are square-free. The method we used in this paper is due to Tolev. [ABSTRACT FROM AUTHOR]

Let χ be a nonprincipal Dirichlet character modulo a prime number p ≽ 3 and let a X : = 1 2 (1 − X (− 1)) . Define the mean value ℳ p (− s , X) : = 2 p − 1 ∑ ψ (mod p) ψ (− 1) = − 1 L (1 , ψ) L (− s , X ψ ¯) (σ : = ℜ s > 0). We give an identity for ℳ p (− s , X) which, in particular, shows that ℳ p (− s , X) = L (1 − s , X) + a X 2 p s L (1 , X) ζ (− s) + o (1) (p → ∞) for fixed 0 < σ < 1 2 and | t : = J s | = o (p (1 − 2 σ) / (3 + 2 σ) ) . [ABSTRACT FROM AUTHOR]

About Lehmer's number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let p be a prime, and let N( k; p) denote the number of all 1 ⩽ a ⩽ p − 1 such that a a ... a ≡ 1 mod p and 2 | a + ā + 1, i = 1, 2, ..., k. The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function N( k; p), and give an interesting asymptotic formula for it. [ABSTRACT FROM AUTHOR]

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 ≡ 1 (mod 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 and give an asymptotic formula by using properties of character sums. [ABSTRACT FROM AUTHOR]

Let q ⩾ 3 be a positive integer. For any integers m and n, the two-term exponential sum C( m, n, k; q) is defined by $C(m,n,k;q) = \sum\limits_{a = 1}^q {e((ma^k + na)/q)} $, where $e(y) = e^{2\pi iy} $. In this paper, we use the properties of Gauss sums and the estimate for Dirichlet character of polynomials to study the mean value problem involving two-term exponential sums and Dirichlet character of polynomials, and give an interesting asymptotic formula for it. [ABSTRACT FROM AUTHOR]