Integers with dense divisors

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are y-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?y. Let D(x,y,z) be the number of positive integers not exceeding x whose divisors are y-dense and whose prime divisors are bigger than z, and let , and . We show that is equivalent, in a large region, to a function d(u,v) which satisfies a difference-differential equation. Using that equation we find that d(u,v)?(1−u/v)/(u+1) for v?3+ε. Finally, we show that d(u,v)=e^−γd(u)+O(1/v), where γ is Euler's constant and d(u)∼x⁻¹D(x,y,1), for fixed u. This leads to a new estimate for d(u).     

On the properties of Newton-Euler pairs

If and are two sequences such that a₁=b₁ and , then we say that (a_n,b_n) is a Newton-Euler pair. In the paper, we establish many formulas for Newton-Euler pairs, and then make use of them to obtain new results concerning some special sequences such as and B_n, where p(n) is the number of partitions of n, σ(n) is the sum of divisors of n, and B_n is the nth Bernoulli number.     

Équidistribution presque partout modulo 1 de suites oscillantes perturbées III : cas liouvillien multidimensionnel

We extend the results of uniform distribution modulo 1 given in [B. Rittaud, Équidistribution presque partout modulo 1 de suites oscillantes perturbées, Bull. Soc. Math. France 128 (2000) 451-471; B. Rittaud, Équidistribution presque partout modulo 1 de suites oscillantes perturbées, II: Cas Liouvillien unidimensionnel, Colloq. Math. 96 (1) (2003) 55-73], which deal with sequences of the form , where n(hn), and are polynomially increasing sequences, n(εn) a bounded sequence, essentially a C³-function Zd-periodic, Θ an element of Rd and t a real number. We remove the Diophantine hypothesis on Θ needed in [the first of above mentioned articles], and add a technical hypothesis on hn. We apply this result to the convergence of diagonal averages for d×d matrices.     

Quartic residues and binary quadratic forms

Let be a prime, m∈Z and . In this paper we obtain a general criterion for m to be a quartic residue in terms of appropriate binary quadratic forms. Let d>1 be a squarefree integer such that , where is the Legendre symbol, and let ε_d be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes p so that ε_d is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where p is an odd prime, and . As an application we also obtain a general criterion for , where {u_n(a,b)} is the Lucas sequence defined by and .     

p-Catalan numbers and squarefree binomial coefficients

In this paper, we consider the generalized Catalan numbers , which we call s-Catalan numbers. For p prime, we find all positive integers n such that p^q divides F(p^q,n), and also determine all distinct residues of , q?1. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. In the second part of the paper we prove that if pq?99999, then is not squarefree for n?τ₁(p^q) sufficiently large (τ₁(p^q) computable). Moreover, using the results of the first part, we find n<τ₁(p^q) (in base p), for which may be squarefree. As consequences, we obtain that is squarefree only for n=1,3,45, and is squarefree only for n=1,4,10.     

Kloosterman's uniformly distributed sequence

Applying the theory of uniform distribution, especially the Erdös-Turán-Koksma inequality and the Koksma-Hlawka inequality, to the two-dimensional Kloosterman sequence , j=1,2,…,?(n) (where , and ?(n) is the Euler function) we find an estimation for the discrepancy of this sequence and an error term for the Kth moment, K=1,2,…, of the sequence of distances as

     

Galois cohomology in degree 3 of function fields of curves over number fields

Let k be a field of characteristic not equal to 2. For n≥1, let denote the nth Galois Cohomology group. The classical Tate's lemma asserts that if k is a number field then given finitely many elements , there exist such that α_i=(a)∪(b_i), where for any λ∈k∗, (λ) denotes the image of k∗ in . In this paper we prove a higher dimensional analogue of the Tate's lemma.     

Quadratic addition rules for quantum integers

For every positive integer n, the quantum integer [n]_q is the polynomial [n]_q=1+q+q²+?+q^n-1. A quadratic addition rule for quantum integers consists of sequences of polynomials , , and such that for all m and n. This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials that satisfy the associated functional equation .     

Positive periodic solutions of functional differential equations

We consider the existence, multiplicity and nonexistence of positive ω-periodic solutions for the periodic equation x′(t)=a(t)g(x)x(t)−λb(t)f(x(t−τ(t))), where are ω-periodic, , , f,g∈C([0,∞),[0,∞)), and f(u)>0 for u>0, g(x) is bounded, τ(t) is a continuous ω-periodic function. Define , , i₀=number of zeros in the set and i_∞=number of infinities in the set . We show that the equation has i₀ or i_∞ positive ω-periodic solution(s) for sufficiently large or small λ>0, respectively.     

On the prime power factorization of n!

In this paper, we prove two results. The first theorem uses a paper of Kim (J. Number Theory 74 (1999) 307) to show that for fixed primes p₁,…,p_k, and for fixed integers m₁,…,m_k, with , the numbers (e_p1(n),…,e_pk(n)) are uniformly distributed modulo (m₁,…,m_k), where e_p(n) is the order of the prime p in the factorization of n!. That implies one of Sander's conjectures from Sander (J. Number Theory 90 (2001) 316) for any set of odd primes. Berend (J. Number Theory 64 (1997) 13) asks to find the fastest growing function f(x) so that for large x and any given finite sequence , there exists n<x such that the congruences hold for all i?f(x). Here, p_i is the ith prime number. In our second result, we are able to show that f(x) can be taken to be at least , with some absolute constant c₁, provided that only the first odd prime numbers are involved.     

A lower bound for the Lindelöf function associated to generalized integers

In this paper we study generalized prime systems for which the integer counting function NP(x) is asymptotically well-behaved, in the sense that NP(x)=ρx+O(xβ), where ρ is a positive constant and . For such systems, the associated zeta function ζP(s) has finite order for , and the Lindelöf function μP(σ) may be defined. We prove that for all such systems, μP(σ)?μ₀(σ) for σ>β, where