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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]

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]

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]

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]

[9] M. Munsch, I. E. Shparlinski and K. H. Yau, 'Smooth squarefree and squarefull integers in arithmeticprogressions', Mathematika66(1)(2020),56-70. 509 510 K.H.Yau [2] proof technique uses the Harman sieve [6] together with the arithmetic information suppliedbyboundsformultiplicativecharactersums. We obtain an asymptotic formula for the number of ways to represent every reduced residue class as a product of a prime and a square-free integer. [Extracted from the article]

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]

Given an elliptic curve $$E$$ over a finite field $$\mathbb {F}_q$$ of $$q$$ elements, we say that an odd prime $$\ell \not \mid q$$ is an Elkies prime for $$E$$ if $$t_E^2 - 4q$$ is a square modulo $$\ell $$ , where $$t_E = q+1 - \#E(\mathbb {F}_q)$$ and $$\#E(\mathbb {F}_q)$$ is the number of $$\mathbb {F}_q$$ -rational points on $$E$$ ; otherwise, $$\ell $$ is called an Atkin prime. We show that there are asymptotically the same number of Atkin and Elkies primes $$\ell < L$$ on average over all curves $$E$$ over $$\mathbb {F}_q$$ , provided that $$L \ge (\log q)^\varepsilon $$ for any fixed $$\varepsilon >0$$ and a sufficiently large $$q$$ . We use this result to design and analyze a fast algorithm to generate random elliptic curves with $$\#E(\mathbb {F}_p)$$ prime, where $$p$$ varies uniformly over primes in a given interval $$[x,2x]$$ . [ABSTRACT FROM AUTHOR]