Igor E. Shparlinski

On the Erdős--Odlyzko--Sárközy conjecture for prime-power moduli

William Banks, Igor E. Shparlinski

Erdős, Odlyzko and Sárközy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$ . O. Klurman, A. P. Mangerel and J. Teräväinen…

Arithmetic structure of $L_{2}$ -norms of $SL_{2} (Z)$ matrices

Igor E. Shparlinski, Yixiu Xiao

For a matrix $γ \in SL_{2} (Z)$ , we define \[ {\mathcal R}(γ)=a_1^2+a_2^2+a_3^2+a_4^2, \qquad \text{where} \ γ=\begin{pmatrix}a_1&a_2\\ a_3&a_4\end{pmatrix}, \] and l…

Large sieve inequality for sums of Legendre symbols over short intervals

Marc Munsch, Igor Shparlinski, Yu-Chen Sun +1

Using the Burgess bound and the Selberg sieve, we obtain an upper bound for the second moment of sums of Legendre symbols over intervals , with the modulus ranging over primes . Th…

Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes

Igor E. Shparlinski, Yixiu Xiao

We establish various upper bounds on Type-I and Type-II shifted bilinear sums with Salié sums modulo a large prime $q$ . We use these bounds to study, for fixed integers $a,b\not \e…

On the Dynamical System Generated by the Möbius Transformation at Smooth Times

László Mérai, Igor E. Shparlinski

We study the distribution of the sequence of the first $N$ elements of the discrete dynamical system generated by the Möbius transformation $x \mapsto (α x + β) / (γ x + δ)$ over a fin…

On the correlations between character sums of division polynomials under shifts

Subham Bhakta, Igor E. Shparlinski

Let $E$ be an elliptic curve over the finite field $F_{p}$ , and $P \in E (F_{p})$ be an $F_{p}$ -rational point. We study the sums \[ S_{χ,P}(N,h) = \sum_{n=1}^N…

On quantum ergodicity for higher dimensional cat maps modulo prime powers

Subham Bhakta, Igor E. Shparlinski

A discrete model of quantum ergodicity of linear maps generated by symplectic matrices $A \in Sp (2 d, Z)$ modulo an integer $N \geq 1$ , has been studied for $d = 1$ and…

Mean value theorems with smooth numbers

F. Majeed, Igor E. Shparlinski

We obtain new mean value theorems for exponential sums with very smooth numbers, which provide a power saving against the trivial bound in region where previous bounds do not apply…

Multiple sums with the Möbius function

William D. Banks, Igor E. Shparlinski

We establish nontrivial bounds for bilinear sums involving the Möbius function evaluated over solutions to a broad class of equations. Several of our results may be regarded as Möb…

Exponential Sums with Sparse Polynomials and Distribution of the Power Generator

Subham Bhakta, Igor Shparlinski

We obtain new bounds on complete rational exponential sums with sparse polynomials modulo a prime, under some mild conditions on the degrees of the monomials of such polynomials. T…

Sums of Kloosterman sums over square-free and smooth integers

Xuancheng Shao, Igor E. Shparlinski, Laurence P. Wijaya

Recently there has been a large number of works on bilinear sums with Kloosterman sums and on sums of Kloosterman sums twisted by arithmetic functions. Motivated by these, we consi…

Binary and ternary congruences involving intervals and sets modulo a prime

Moubariz Z. Garaev, Julio C. Pardo, Igor E. Shparlinski

Let $s$ be a fixed positive integer constant, $ε$ be a fixed small positive number. Then, provided that a prime $p$ is large enough, we prove that for any set $\{{\mathca…