On the product of two primitive elements of maximal subfields of a finite field

Let denote a finite field with r elements. Let q be a power of a prime, and p₁,p₂, p₃ be distinct primes. Put

     

Notes sur la fonction ζ de Riemann, 4

Opération fondamentale de l'arithmétique, familière depuis des millénaires, la division euclidienne n'a pas livré tous ses secrets. Ainsi, notons pour k et a entiers positifs, le reste de la division euclidienne de k par a, et imaginons un instant que, par un choix convenable d'un entier n et de réels c₂,…,c_n, nous sachions rendre arbitrairement petite la quantité

     

On character sums over flat numbers

Let q?2 be an integer, χ be any non-principal character mod q, and H=H(q)?q. In this paper the authors prove some estimates for character sums of the form

     

On sums of binomial coefficients and their applications

In this paper we study recurrences concerning the combinatorial sum and the alternate sum , where m>0, n?0 and r are integers. For example, we show that if n?m-1 then

     

On consecutive happy numbers

Let e?1 and b?2 be integers. For a positive integer with 0?aj<b, define

     

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

We study spectral properties for HK,Ω, the Krein-von Neumann extension of the perturbed Laplacian −Δ+V defined on , where V is measurable, bounded and nonnegative, in a bounded open set Ω⊂Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r>1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula

     

On the parity of exponents in the standard factorization of n!

Let p₁,p₂,… be the sequence of all primes in ascending order. The following result is proved: for any given positive integer k and any given , there exist infinitely many positive integers n with

     

On various restricted sumsets

For finite subsets A₁,…,A_n of a field, their sumset is given by . In this paper, we study various restricted sumsets of A₁,…,A_n with restrictions of the following forms:

     

Openness of the Galois image for τ-modules of dimension 1

Let C be a smooth projective absolutely irreducible curve over a finite field , F its function field and A the subring of F of functions which are regular outside a fixed point ∞ of C. For every place ? of A, we denote the completion of A at ? by .In [Pi2], Pink proved the Mumford-Tate conjecture for Drinfeld modules. Let φ be a Drinfeld module of rank r defined over a finitely generated field K containing F. For every place ? of A, we denote by Γ_? the image of the representation

     

On certain exponential sums over primes

Let f(x) be a real valued polynomial in x of degree k?4 with leading coefficient α. In this paper, we prove a non-trivial upper bound for the quantity

     

A combinatorial identity with application to Catalan numbers

By a very simple argument, we prove that if l,m,n∈{0,1,2,…} then

     

Note on some congruences of Lehmer

In the paper, we generalize some congruences of Lehmer and prove that for any positive integer n with (n,6)=1

     

On the residue classes of integer sequences satisfying a linear three term recurrence formula

In this paper we investigate linear three-term recurrence formulae with sequences of integers (T(n))_n?0 and (U(n))_n?0, which are ultimately periodic modulo m, e.g.

     

On the independence of Heegner points associated to distinct quadratic imaginary fields

Let E/Q be an elliptic curve with no CM and a fixed modular parametrization and let be Heegner points attached to the rings of integers of distinct quadratic imaginary fields k₁,…,kr. We prove that if the odd parts of the class numbers of k₁,…,kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in .     

A formula for the central value of certain Hecke L-functions

Let be the negative of a prime, and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to . Under these assumptions, there exists Hecke characters ψ_D of K with conductor (D) and infinite type (1,0). Their L-series L(ψ_D,s) are associated to a CM elliptic curve A(N,D) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L(ψ_D,s) of the form L(ψ_D,1)=Ω∑_[A],Ir(D,[A],I)m_[A],I([D]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [A] are class group representatives of K. An application of this formula for the case N=-7 will allow us to prove the non-vanishing of a family of L-series of level 7|D| over K.     

Congruences involving Bernoulli and Euler numbers

Let [x] be the integral part of x. Let p>5 be a prime. In the paper we mainly determine , , and in terms of Euler and Bernoulli numbers. For example, we have

     

Explicit upper bounds for values at s=1 of Dirichlet L-series associated with primitive even characters

Let S be a given finite set of pairwise distinct rational primes. We give an explicit constant κ_S such that for any even primitive Dirichlet character χ of conductor q_χ>1 we have

     

On random perturbation of some intersective sets

Given a subset S of Z and a sequence I = (I_n)_n=1^∞ of intervals of increasing length contained in Z, let