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The primary contribution of this paper is to research one kind special Kloosterman sums using analytic method. We translate one kind fourth power mean of the Kloosterman sums into the character sums of one kind polynomials. It is possible to construct an exact formula for the fourth power mean of one kind Kloosterman sums by computing the character sums of the polynomials. The paper is novel because of its proof and conversion strategies. [ABSTRACT FROM AUTHOR]

Let α be an irrational number and let β be a real number. The corresponding Beatty sequence is defined as B α , β = { ⌊ α n + β ⌋ : n ∈ N } , where ⌊ y ⌋ is the largest integer not exceeding y. We study the distribution of square-free primitive roots in the Beatty sequence B α , β . As an intermediate step, bounds for character sums with square-free numbers in Beatty sequences are obtained. [ABSTRACT FROM AUTHOR]

Diaconis and Gamburd computed moments of secular coefficients in the CUE ensemble. We use the characteristic map to give a new combinatorial proof of their result. We also extend their computation to moments of traces of symmetric powers, where the same result holds but in a wider range. Our combinatorial proof is inspired by gcd matrices, as used by Vaughan and Wooley and by Granville and Soundararajan. We use these CUE computations to suggest a conjecture about moments of characters sums twisted by the Liouville (or by the Möbius) function, and establish a version of it in function fields. The moral of our conjecture (and its verification in function fields) is that the Steinhaus random multiplicative function is a good model for the Liouville (or for the Möbius) function twisted by a random Dirichlet character. We also evaluate moments of secular coefficients and traces of symmetric powers, without any condition on the size of the matrix. As an application we give a new formula for a matrix integral that was considered by Keating, Rodgers, Roditty-Gershon and Rudnick in their study of the k-fold divisor function. [ABSTRACT FROM AUTHOR]

Let p be a fixed odd prime and Q (x , y , z) = a x 2 + b x y + c y 2 + d x z + e y z + f z 2 be a fixed quadratic form in Z [ x , y , z ] which is non-degenerate in F p [ x , y , z ] and gcd (a (4 a c - b 2) , p) = 1. Let (x 0 , y 0 , z 0) be a fixed point in Z 3 . We study the behavior of solutions (x, y, z) of congruences of the form Q (x , y , z) ≡ 0 mod q with q = p n , where max { | x - x 0 | , | y - y 0 | , | z - z 0 | } ≤ N and gcd (z , p) = 1. In fact, we consider a smooth version of this problem and establish an asymptotic formula (thus the existence of such solutions) when n → ∞ , under the condition N ≥ q 1 2 + ε . [ABSTRACT FROM AUTHOR]

The main purpose of this article is to use elementary methods and properties of classical Gauss sums to determine identities for the number of residue systems of aa mod pp such that aa, a+a¯a+\bar{a}, and a−a¯a-\bar{a} are all quadratic residues, improving on results of Wang and Lv.

We estimate weighted character sums with determinants ad − bc of 2 × 2 matrices modulo a prime p with entries a, b, c and d varying over the interval [1, N]. Our goal is to obtain non-trivial bounds for values of N as small as possible. In particular, we achieve this goal, with a power saving, for N ⩾ p1/8+ε with any fixed ε > 0, which is very likely to be the best possible unless the celebrated Burgess bound is improved. By other techniques, we also treat more general sums but sometimes for larger values of N. [ABSTRACT FROM AUTHOR]

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Pólya–Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess’ estimate for short character sums, and upper bounds for L (1 , χ) and L (1 + i t , χ) ) are more-or-less “equivalent”. We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions. [ABSTRACT FROM AUTHOR]

We study averages involving the Jacobi quadratic symbol (n m) in regions where the product mn is bounded by a large parameter. We show that these averages exhibit cancellation whenever the summation is restricted to square-free integers bounded away from the axes. [ABSTRACT FROM AUTHOR]

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]

In this paper, we present a new reciprocity formula for certain Hardy sums and study a computational problem of one type mean value containing certain Hardy sums and the two-term exponential sum with the help of the properties of Gauss sums and Dirichlet L - function. [ABSTRACT FROM AUTHOR]

In this paper we characterize the c-Boomerang Connectivity Table (BCT), c ≠ 0 (thus, including the classical c = 1 case), for all monomial function x d in terms of characters and Weil sums on the finite field F p n , for an odd prime p. We further simplify these expressions for the Gold functions x p k + 1 for all 1 ≤ k < n , and p odd. It is the first such attempt for a complete description for the classical BCT and its relative c-BCT, for all parameters involved, albeit in terms of characters. [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]

An asymptotic formula for the number of positive primitive roots, which are r-full integers and not exceeding a given range, is derived using properties of character sums and exponent pairs. [ABSTRACT FROM AUTHOR]

We provide examples of multiplicative functions f supported on the k-free integers such that at primes f (p) = ± 1 and such that the partial sums of f up to x are o (x 1 / k) . Further, if we assume the Generalized Riemann Hypothesis, then we can improve the exponent 1/k: There are examples such that the partial sums up to x are o (x 1 / (k + 1 2) + ϵ) , for all ϵ > 0 . This generalizes to the k-free integers the results of Aymone (J. Number Theory, 212:113-121, 2020). [ABSTRACT FROM AUTHOR]

Let q ≥ 2 be an integer and let s q (n) be the sum-of-digitsfunction of n in base q. The function s q (n) has been studied in many directions and many properties have been obtained on the distribution of s q (n) and s q (P (n)) , where P is a suitable polynomial. In this paper we derive the generatingfunctions of s p (n d mod p k) for d ≥ 2 and prime p ≥ 2 by using the properties of Dirichlet character, and study the correlation properties of the sequences ((- 1) n 2 mod p k ) n < p k . [ABSTRACT FROM AUTHOR]

Let p > 5 be a fixed prime and assume that α 1 , α 2 , α 3 are coprime to p. We study the asymptotic behavior of small solutions of congruences of the form α 1 x 1 2 + α 2 x 2 2 + α 3 x 3 2 ≡ 0 mod q with q = p n , where max { | x 1 | , | x 2 | , | x 3 | } ≤ N and (x 1 x 2 x 3 , p) = 1 . (In fact, we consider a smoothed version of this problem.) If α 1 , α 2 , α 3 are fixed and n → ∞ , we establish an asymptotic formula (and thereby the existence of such solutions) under the condition N ≫ q 1 / 2 + ε . If these coefficients are allowed to vary with n, we show that this formula holds if N ≫ q 11 / 18 + ε . The latter should be compared with a result by Heath-Brown who established the existence of non-zero solutions under the condition N ≫ q 5 / 8 + ε for odd square-free moduli q. [ABSTRACT FROM AUTHOR]

In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27\ldots \%$ of the L-functions under consideration do not vanish at 1/2. [ABSTRACT FROM AUTHOR]

We consider the problem of bounding the dimension of Hilbert cubes in a finite field F p that does not contain any primitive roots. We show that the dimension of such Hilbert cubes is O ε (p 1 / 8 + ε) for any ε > 0 , matching what can be deduced from the classical Burgess estimate in the special case when the Hilbert cube is an arithmetic progression. We also consider the dual problem of bounding the dimension of multiplicative Hilbert cubes avoiding an interval. [ABSTRACT FROM AUTHOR]

We give an effective version with explicit constants of the large sieve inequality for imaginary quadratic fields. Explicit results of this kind are useful for estimating the computational complexity of algorithms which generate elements, whose norm is a rational prime, in an arithmetic progression of the corresponding ring of integers.

In a recent paper devoted to fault analysis of elliptic curve-based signature schemes, Takahashi et al. (TCHES 2018) described several attacks, one of which assumed an equidistribution property that can be informally stated as follows: given an elliptic curve E over 𝔽q in Weierstrass form and a large subgroup H ⊂ E(𝔽q) generated by G(xG, yG), the points in E(𝔽q) whose x-coordinates are obtained from xG by randomly flipping a fixed, sufficiently long substring of bits (and rejecting cases when the resulting value does not correspond to a point in E(𝔽q)) are close to uniformly distributed among the cosets modulo H. The goal of this note is to formally state, prove and quantify (a variant of) that property, and in particular establish sufficient bounds on the size of the subgroup and on the length of the substring of bits for it to hold. The proof relies on bounds for character sums on elliptic curves established by Kohel and Shparlinski (ANTS–IV).

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]

Let K be a cyclic number field of odd degree over Q with odd narrow class number, such that 2 is inert in K / Q . We define a family of number fields { K (p) } p , depending on K and indexed by the rational primes p that split completely in K / Q , in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in K (p) / Q is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension K / Q . Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals. [ABSTRACT FROM AUTHOR]

Dartyge and Sárközy (partly with other coauthors) introduced pseudorandom measures of subsets, and studied the pseudorandomness of certain subsets related to primitive roots. In this paper we answer a conjecture of Dartyge, Sárközy and Szalay, and study the pseudorandom properties of some subsets constructed by using primitive roots. [ABSTRACT FROM AUTHOR]

We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. Our best results is conditional on the Generalised Riemann hypothesis (GRH). As a corollary, we make progress on a conjecture of Erdös, Odlyzko, and Sárközy. [ABSTRACT FROM AUTHOR]

Let q ≥ 2 be an integer and let s q (n) be the sum-of-digits function in base q of the positive integer n. In this paper we obtain asymptotic formulae for the distribution of (s q (n 2 mod q k)) n < q k and (s p (n d mod p k)) n < p k in residue classes modulo m, where q ≥ 2 , m ≥ 2 and d ≥ 2 are general integers, p > 2 is a prime. Furthermore, we give exact identities for the distribution of (s p (n d mod p k)) n < p k in residue classes modulo p. The properties of Dirichlet character sums and exponential sums play an important role in the proof of the results. [ABSTRACT FROM AUTHOR]

In this survey, we explore how superorthogonality amongst functions in a sequence f 1 , f 2 , f 3 , ... results in direct or converse inequalities for an associated square function. We distinguish between three main types of superorthogonality, which we demonstrate arise in a wide array of settings in harmonic analysis and number theory. This perspective gives clean proofs of central results, and unifies topics including Khintchine's inequality, Walsh–Paley series, discrete operators, decoupling, counting solutions to systems of Diophantine equations, multicorrelation of trace functions, and the Burgess bound for short character sums. [ABSTRACT FROM AUTHOR]

Using ℓ-adic cohomology of tensor inductions of lisse Qℓ-sheaves, we study a class of incomplete character sums.

The main aim of this paper is to use the analytic methods and the properties of the classical Gauss sums to research the computational problem of one kind hybrid power mean containing the character sums of polynomials and two-term exponential sums modulo p, an odd prime, and acquire several accurate asymptotic formulas for them.