On the distribution of the distances of multiples of an irrational number to the nearest integer
On the divisibility properties of integers (I)
On the divisibility properties of sequences of integers (II)
On the divisor function over Piatetski-Shapiro sequences
Let be an integer part of and be the number of positive divisor of . Inspired by some results of M. Jutila (1987), we prove that for , where is the Euler constant and is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.
On the divisors of
On the error term of the logarithm of the lcm of a quadratic sequence
We study the logarithm of the least common multiple of the sequence of integers given by . Using a result of Homma [5] on the distribution of roots of quadratic polynomials modulo primes we calculate the error term for the asymptotics obtained by Cilleruelo [3].
On the existence of a density
On the Fermat periods of natural numbers.
On the height of cyclotomic polynomials
On the Lehmer constant of finite cyclic groups
On the measure of measurable sets of integers
On the Multiplicative Group Generated by a Dense Sequence of Integers.
On the (non-)existence of m-cycles for generalized Syracuse sequences
On the non-holonomic character of logarithms, powers, and the th prime function.
On the product of balanced sequences
The product w = u ⊗ v of two sequences u and v is a naturally defined sequence on the alphabet of pairs of symbols. Here, we study when the product w of two balanced sequences u,v is balanced too. In the case u and v are binary sequences, we prove, as a main result, that, if such a product w is balanced and deg(w) = 4, then w is an ultimately periodic sequence of a very special form. The case of arbitrary alphabets is approached in the last section. The partial results obtained and the problems...
On the product of balanced sequences
The product w = u ⊗ v of two sequences u and v is a naturally defined sequence on the alphabet of pairs of symbols. Here, we study when the product w of two balanced sequences u,v is balanced too. In the case u and v are binary sequences, we prove, as a main result, that, if such a product w is balanced and deg(w) = 4, then w is an ultimately periodic sequence of a very special form. The case of arbitrary alphabets is approached in the last section. The partial results obtained and the problems...
On the roots of certain sequences of congruences
On the structure of sets with small doubling property on the plane (I)
Let K be a finite set of lattice points in a plane. We prove that if |K| is sufficiently large and |K+K| < (4 - 2/s)|K| - (2s-1), then there exist s - 1 parallel lines which cover K. We also obtain some more precise structure theorems for the cases s = 3 and s = 4.
On the upper asymptotic density of (0, r)-primitive sequences
On the function.