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On the Euler Function on Differences Between the Coordinates of Points on Modular Hyperbolas

100%

Shparlinski I.

nr 1

1-7

For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola $𝓗_{a,p}(X,Y) = {(x,y) : xy ≡ a(mod p), 1 ≤x≤X, 1 ≤y≤Y}$. We give asymptotic formulas for the average values $∑_{\substack (x,y)∈ 𝓗_{a,p}(X,Y) x ≠ y*} φ(|x-y|)/|x-y|$ and $∑_{\substack (x,y)∈ 𝓗_{a,p}(X,X) x ≠ y*} φ(|x-y|)$ with the Euler function φ(k) on the differences between the components of points of $𝓗_{a,p}(X,Y)$.

Exponential Sums with Farey Fractions

100%

Shparlinski I.

nr 2

101-107

For positive integers m and N, we estimate the rational exponential sums with denominator m over the reductions modulo m of elements of the set ℱ(N) = {s/r : r,s ∈ ℤ, gcd(r,s) = 1, N ≥ r > s ≥ 1} of Farey fractions of order N (only fractions s/r with gcd(r,m) = 1 are considered).

Twisted exponential sums over points of elliptic curves

63%

Ostafe A. , Shparlinski I.

tom 148

nr 1

77-92

Distribution of consecutive modular roots of an integer

63%

Bourgain J. , Shparlinski I.

tom 134

nr 1

83-91

Prime numbers with Beatty sequences

63%

Banks W. , Shparlinski I.

Colloquium Mathematicum

2009

tom 115

nr 2

147-157

A study of certain Hamiltonian systems has led Y. Long to conjecture the existence of infinitely many primes which are not of the form p = 2⌊αn⌋ + 1, where 1 < α < 2 is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately implies that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic formula for the number of primes p = q⌊αn + β⌋ + a with n ≤ N, where α,β are real numbers such that α is positive and irrational of finite type (which is true for almost all α) and a,q are integers with $0 ≤ a < q ≤ N^κ$ and gcd(a,q) = 1, where κ > 0 depends only on α. We also prove a similar result for primes p = ⌊αn + β⌋ such that p ≡ a(mod q).

On a ternary quadratic form over primes

63%

Fouvry É. , Shparlinski I.

tom 150

nr 3

285-314

Arithmetic properties of numbers with restricted digits

63%

Banks W. , Shparlinski I.

tom 112

nr 4

313-332

Visible Points on Modular Exponential Curves

63%

Chan T. , Shparlinski I.

tom 58

nr 1

17-22

We obtain an asymptotic formula for the number of visible points (x,y), that is, with gcd(x,y) = 1, which lie in the box [1,U] × [1,V] and also belong to the exponential modular curves $y ≡ ag^{x} (mod p)$. Among other tools, some recent results of additive combinatorics due to J. Bourgain and M. Z. Garaev play a crucial role in our argument.

Average multiplicative orders of elements modulo n

63%

Luca F. , Shparlinski I.

nr 4

387-411

On the number of solutions of exponential congruences

51%

Balog A. , Broughan K. , Shparlinski I.

tom 148

nr 1

93-103

On the smallest pseudopower

51%

Bourgain J. , Konyagin S. , Pomerance C. , Shparlinski I.

tom 140

nr 1

43-55