The degree of primitive sequences and Erdős conjecture

A sequence A of strictly positive integers is said to be primitive if none of its term divides another. Z. Zhang proved a result, conjectured by Erdős and Zhang in 1993, on the primitive sequences whose the number of the prime factors of its terms counted with multiplicity is at most 4. In this paper, we extend this result to the primitive sequences whose the number of the prime factors of its terms counted with multiplicity is at most 5.

Distribution and non-vanishing of special values of L-series attached to Erdős functions

In a written correspondence with A. Livingston, Erdős conjectured that for any arithmetical function [Formula: see text], periodic with period [Formula: see text], taking values in [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text], the series [Formula: see text] does not vanish. This conjecture is still open in the case [Formula: see text] or when [Formula: see text]. In this paper, we obtain the characteristic function of the limiting distribution of [Formula: see text] for any positive integer [Formula: see text] and Erdős function [Formula: see text] with the same parity as [Formula: see text]. Moreover, we show that the Erdős conjecture is true with “probability” one.

On a Conjecture of Livingston

In an attempt to resolve a folklore conjecture of Erdös regarding the non-vanishing at s = 1 of the L-series attached to a periodic arithmetical function with period q and values in {−1, 1}, Livingston conjectured the -linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston’s conjecture for composite q ≥ 4, highlighting that a newapproach is required to settle Erdös conjecture. We also prove that the conjecture is true for prime q ≥ 3, and indicate that more ingredients will be needed to settle Erdös conjecture for prime q.

Fundamental units and consecutive squarefull numbers

We connect several seemingly unrelated conjectures of Ankeny, Artin, Chowla and Mordell to a conjecture of Erdös on consecutive squarefull numbers. We then study the Erdös conjecture and relate it to the abc conjecture. We also derive by elementary methods several unconditional results pertaining to the Erdös conjecture.

On a Sumset Conjecture of Erdős

Erdős conjectured that for any set A ⊆ ℕ with positive lower asymptotic density, there are infinite sets B;C ⊆ ℕ such that B + C ⊆ A. We verify Erdős’ conjecture in the case where A has Banach density exceeding ½ . As a consequence, we prove that, for A ⊆ ℕ with positive Banach density (amuch weaker assumption than positive lower density), we can find infinite B;C ⊆ ℕ such that B+C is contained in the union of A and a translate of A. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős’ conjecture for subsets of the natural numbers that are pseudorandom.