Values of linear recurring sequences of vectors over finite fields
On the number of zero trace elements in polynomial bases for F.
Let F denote the finite field of q elements. O. Ahmadi and A. Menezes have recently considered the question about the possible number of elements with zero trace in polynomial bases of F over F. Here we show that the Weil bound implies that there is such a basis with n + O(log n) zero-trace elements.
On the Euler Function on Differences Between the Coordinates of Points on Modular Hyperbolas
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 . We give asymptotic formulas for the average values and with the Euler function φ(k) on the differences between the components of points of .
Primitive Points on a Modular Hyperbola
For positive integers m, U and V, we obtain an asymptotic formula for the number of integer points (u,v) ∈ [1,U] × [1,V] which belong to the modular hyperbola uv ≡ 1 (mod m) and also have gcd(u,v) =1, which are also known as primitive points. Such points have a nice geometric interpretation as points on the modular hyperbola which are "visible" from the origin.
Exponential Sums with Farey Fractions
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).
Small discriminants of complex multiplication fields of elliptic curves over finite fields
We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves over a prime finite field of elements, such that the discriminant of the quadratic number field containing the endomorphism ring of over is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).
On the Győry-Sárközy-Stewart conjecture in function fields
We consider function field analogues of the conjecture of Győry, Sárközy and Stewart (1996) on the greatest prime divisor of the product for distinct positive integers , and . In particular, we show that, under some natural conditions on rational functions , the number of distinct zeros and poles of the shifted products and grows linearly with if . We also obtain a version of this result for rational functions over a finite field.
On the sum of iterations of the Euler function.
On the set of distances between two sets over finite fields.
Distribution of roots of polynomial congruences.
Exponential sums and the distribution of inversive congruential pseudorandom numbers with prime-power modulus
On the concentration of points on modular hyperbolas and exponential curves
Twisted exponential sums over points of elliptic curves
On a ternary quadratic form over primes
On the largest prime factor of
For an integer we denote by the largest prime factor of . We obtain several upper bounds on the number of solutions of congruences of the form and use these bounds to show that
Average multiplicative orders of elements modulo n
Distribution of consecutive modular roots of an integer
Arithmetic properties of numbers with restricted digits
Prime numbers with Beatty sequences
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...
On the distribution of the Euler function of shifted smooth numbers
We give asymptotic formulas for some average values of the Euler function on shifted smooth numbers. The result is based on various estimates on the distribution of smooth numbers in arithmetic progressions which are due to A. Granville and É. Fouvry & G. Tenenbaum.
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