International Journal of Number Theory

We give a new proof of Milne's formulas for the number of representations of an integer as a sum of 4m² and 4m(m + 1) squares. The proof is based on explicit evaluation of pfaffians with elliptic function entries, and relates Milne's formulas to Schur Q-polynomials and to correlation functions for continuous dual Hahn polynomials. We also state a new formula for 2m² squares.

We give bounds on the number of integers 1 ≤ n ≤ N such that p | s(n), where p is a prime and s(n) is the sum of aliquot divisors function given by s(n) = σ(n) - n, where σ(n) is the sum of divisors function. Using this result, we obtain nontrivial bounds in certain ranges for rational exponential sums of the form

We define p-adic q-Bernoulli numbers by using a p-adic integral. These numbers have good properties similar to those of the classical Bernoulli numbers. In particular, they satisfy an analogue of the von Staudt–Clausen theorem, which includes information of denominators of p-adic q-Bernoulli numbers.

Set addition theory is born a few decades ago from additive number theory. Several difficult issues, more combinatorial in nature than algebraic, have been revealed. In particular, computing the values taken by the function:

Recently, a method called isoperimetric, has been developed by Hamidoune and allowed new proofs and generalizations of the classical theorems in additive number theory. For instance, a new interpretation of the isoperimetric method has been able to give a new proof of Kneser's Theorem.

The purpose of this article is to adapt this last proof in a non-abelian group, in order to give new values of the function μ_G, for some solvable groups and alternating groups. These values allow us in particular to answer negatively a question asked in the literature on the μ_G functions.

Fix an integer m > 1, and let denote the group of reduced residues modulo m and set ζ_m = exp(2 πi/m). Let denote the multiplicative inverse of x modulo m. The n-dimensional Kloosterman sums are defined by

In this note, we find all the solutions of the Diophantine equation x² + 2^a · 5^b = yⁿ in positive integers x, y, a, b, n with x and y coprime and n ≥ 3.

We apply the Hecke operators T(p)² and (1 ≤ j ≤ n ≤ 2k) to a degree n theta series attached to a rank 2k ℤ-lattice L equipped with a positive definite quadratic form in the case that L/pL is regular. We explicitly realize the image of the theta series under these Hecke operators as a sum of theta series attached to certain sublattices of , thereby generalizing the Eichler Commutation Relation. We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these operators. We explicitly compute the eigenvalues on the average theta series, extending previous work where we had the restrictions that χ(p) = 1 and n ≤ k. We also show that for j > k when χ(p) = 1, and for j ≥ k when χ(p) = -1, and that θ(gen L) is an eigenform for T(p)².

Nonlinear congruential pseudorandom number generators can have unexpectedly short periods. Shamir and Tsaban introduced the class of counter-dependent generators which admit much longer periods. In this paper, using a technique developed by Niederreiter and Shparlinski, we present discrepancy bounds for sequences of s-tuples of successive pseudorandom numbers generated by counter-dependent generators modulo a composite M.

Let K = ℚ(θ) be an algebraic number field with θ in the ring A_K of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ℚ of rational numbers. For a rational prime p, let be the factorization of the polynomial obtained by replacing each coefficient of f(x) modulo p into product of powers of distinct monic irreducible polynomials over ℤ/pℤ. Dedekind proved that if p does not divide [A_K : ℤ[θ]], then the factorization of pA_K as a product of powers of distinct prime ideals is given by , with 𝔭_i = pA_K + g_i(θ)A_K, and residual degree . In this paper, we prove that if the factorization of a rational prime p in A_K satisfies the above-mentioned three properties, then p does not divide [A_K:ℤ[θ]]. Indeed the analogue of the converse is proved for general Dedekind domains. The method of proof leads to a generalization of one more result of Dedekind which characterizes all rational primes p dividing the index of K.

In 1920, Ramanujan wrote to Hardy about his discovery of the mock theta functions. In the years since, there has been much work in understanding the transformation properties and asymptotic nature of these functions. Recently, Zwegers proved a relationship between mock theta functions and vector-valued modular forms, and Bringmann and Ono used the theory of Maass forms and Poincaré series to prove a conjecture of Andrews, yielding an exact formula for the coefficients of the f(q) mock theta function. Here we build upon these results, using the theory of vector-valued modular forms and Poincaré series to prove an exact formula for the coefficients of the ω(q) mock theta function.

Let ϕ be a Drinfel'd module defined over a finite extension K of 𝔽_q(T); we establish a uniform lower bound for the canonical height of a point of ϕ, rational over the maximal abelian extension of K, and thus solve the so-called abelian version of the Lehmer problem in this situation. The classical original problem (a non torsion point in 𝔾_m(ℚ^ab)) was solved by Amoroso and Dvornicich [1].

Soit ϕ un module de Drinfel'd défini sur une extension finie K de 𝔽_q(T); nous démontrons une minoration uniforme pour la hauteur canonique d'un point de ϕ, rationnel sur l'extension abélienne maximale de K, et résolvons ainsi la version dite abélienne du problème de Lehmer dans cette situation. Dans le cadre classique (un point d'ordre infini de 𝔾_m(ℚ^ab)), cette question a été résolue par Amoroso et Dvornicich dans [1].