35 citations · 42 across the 15 of their papers we have counts for
19 papers · 1 filter
Short sums of the Möbius function over function fields
Etienne Fouvry, Igor E. Shparlinski, Ping Xi
We develop a new method which allows us to estimate, with a power saving, sums of the M{ö}bius function over function fields . The sums run over monic polynomials…
Systems of equations with elements of finitely generated groups and with linear recurrence sequences
Subham Bhakta, Alina Ostafe, Igor E. Shparlinski
We obtain new and essentially optimal bounds on the number of solutions of diagonal systems of polynomial equations in variables from an arbitrary finite subset of a finite…
Tridimensional character sums with polynomial arguments and applications
Étienne Fouvry, Igor E. Shparlinski, Ping Xi
Let be a large prime and a non-trivial Dirichlet character modulo . We study the character sum \[ \sum_{a \sim A} \sum_{b \sim B} \sum_{c \sim C} α(a,b) β(c)χ(f(a) + bc)…
Counting consecutive multiplicatively dependent triples
Igor Shparlinski, Nicolas Sleiman
We obtain a nontrivial bound on the number of triples of positive integers , which are multiplicatively dependent but each pair of its elements is not, and so…
On the generation of multiplicative groups by small primes
Oleksiy Klurman, Igor E. Shparlinski, Joni Teräväinen
Motivated by a question of Regev arising from his improved quantum factoring algorithm, we study how many small primes are needed to generate the group $({\mathbb Z}/q{\mathbb Z})^…
Counting Irreducible polynomials with coefficients from thin subgroups
Alina Ostafe, Igor E. Shparlinski
L. Bary-Soroker and R. Shmueli (2026) have given an asymptotic formula for the number of irreducible polynomials over the finite fields of elements, such that the…
Multiplicative dependence in the denominators of points of elliptic curves
Attila Bérczes, Subham Bhakta, Lajos Hajdu +2
Let be , not necessary distinct, elliptic curves over . We give upper bounds on the frequency of -tuples of points in $E_1(\mathbb{Q})\times \…
Counting solvable -unit equations and linear recurrence sequences with zeros
Alina Ostafe, Carl Pomerance, Igor E. Shparlinski
We show that only a rather small proportion of linear equations are solvable in elements of a fixed finitely generated subgroup of a multiplicative group of a number field. The arg…
Counting integer matrices with square-free determinants
Alina Ostafe, Igor E. Shparlinski
We consider the set of -matrices with integer elements of size at most and obtain an asymptotic formula for the number of…
Counting matrices over finite rank multiplicative groups
Aaron Manning, Alina Ostafe, Igor E. Shparlinski
Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets of finite rank…
Multiplicative dependence in linear recurrence sequences
Attila Bérczes, Lajos Hajdu, Alina Ostafe +1
For a wide class of integer linear recurrence sequences , we give an upper bound on the number of -tuples $\left(n_1, \ldots, n_s\right) \in \lef…
On Artin's conjecture on average and short character sums
Oleksiy Klurman, Igor E. Shparlinski, Joni Teräväinen
Let denote the number of primes up to for which the integer is a primitive root. We show that satisfies the asymptotic predicted by Artin's conjecture for…
On the determinants of matrices with elements from arbitrary sets
Ilya D. Shkredov, Igor E. Shparlinski
Recently there has been several works estimating the number of matrices with elements from some finite sets of arithmetic interest and of a given determina…
On the frequency of primes preserving dynamical irreducibility of polynomials
Alina Ostafe, Igor E. Shparlinski
Towards a well-known open question in arithmetic dynamics, L. Mérai, A. Ostafe and I. E. Shparlinski (2023), have shown, for a class of polynomials , which in p…
Character sums over elements of extensions of finite fields with restricted coordinates
Siddharth Iyer, Igor Shparlinski
We obtain nontrivial bounds for character sums with multiplicative and additive characters over finite fields over elements with restricted coordinate expansion. In particular, we…
Counting elements of the congruence subgroup
Kamil Bulinski, Igor E. Shparlinski
We obtain asymptotic formulas for the number of matrices in the congruence subgroup \[ Γ_0(Q) = \left\{ A\in\mathrm{SL}_2(\mathbb Z):~c \equiv 0 \pmod Q\right\}, \] which are of na…
Bounds on bilinear forms with Kloosterman sums
Bryce Kerr, Igor E. Shparlinski, Xiaosheng Wu +1
We prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by É. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, É. Fouvry,…
On smooth square-free numbers in arithmetic progressions
Marc Munsch, Igor E. Shparlinski
A. Booker and C. Pomerance (2017) have shown that any residue class modulo a prime can be represented by a positive -smooth square-free integer wit…
Fermat quotients: Exponential sums, value set and primitive roots
Igor E. Shparlinski
For a prime and an integer with , we define Fermat quotients by the conditions D…