Abstract. We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GLpNq Maass cusp forms for all N > 2, satisfy a central limit theorem in a suitable range, generalizing the case N “ 2 treated by É. Fouvry, S. Ganguly, E. Kowalski and P. Michel in [4]. Such universal Gaussian behaviour relies on a deep e...

d|en d denote the number and the sum of exponential divisors of n, respectively. Properties of these functions were investigated by several authors, see [1], [2], [3], [5], [6], [8]. Two integers n,m > 1 have common exponential divisors iff they have the same prime factors and for n = ∏r i=1 p ai i , m = ∏r i=1 p bi i , ai, bi ≥ 1 (1 ≤ i ≤ r), the greatest common exponential divisor of n and m is

As usual Z, Q, R and C denote the ring of integers, the rational field, the real field and the complex field respectively. We also let Z = {1, 2, 3, · · · } and C∗ = C \ {0}. For a ∈ Z and n ∈ Z, by (a, n) we mean th greatest common divisor of a and n, if n is odd then the Jacobi symbol ( a n ) is defined in terms of Legendre symbols (see, e.g. [IR]). For x ∈ R, [x] and {x} stand for the integr...

Let p be a prime, q a prime divisor in p−1, and g an element of order q in Z∗ p . Suppose a prover P has chosen w in Zq at random and has published h = gw mod p. A verifier V who gets p, q, g, h can check that p, q are prime, and that g, h have order q. Since there is only one subgroup of order q in Z∗ p , this automatically means that h ∈< g >, i.e. there exists w such that h = gw. But this do...

“Never underestimate a theorem that counts something!” – or so says J. Fraleigh in his classic text [2]. Indeed, in [1] and [4], the authors derive Fermat’s (little), Lucas’s and Wilson’s theorems, among other results, all from a single combinatorial lemma. This lemma can be derived by applying Burnside’s theorem to an action by a cyclic group of prime order. In this note, we generalize this le...

Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is said to be perfect if σ(n) = 2n. It is well known that a number is even and perfect if and only if it has the form 2p−1(2p − 1) where 2p − 1 is prime. No odd perfect numbers are known, nor has any proof of their nonexistence ever been given. In the meantime, much work has been done in establishing conditions ...

Let p be a prime number, and F p the nite eld with p elements. Let S(m) be the \smoothness" function that for integers m is deened as the largest prime divisor of m. In this note, we prove the following theorem. Theorem. There is a deterministic algorithm for factoring polynomials over F p , which on poly-nomials over F p of degree n runs in time S(p ? 1) 1=2 (n log p) O(1) under the assumption...

Let q > 1 denote an integer relatively prime to 2, 3, 7 and for which G = PSL(2, q) is a Hurwitz group for a smooth projective curve X defined over C. We compute the G-module structure of the RiemannRoch space L(D), where D is an invariant divisor on X of positive degree. This depends on a computation of the ramification module, which we give explicitly. In particular, we obtain the decompositi...

We present a modular algorithm for computing GCDs of univariate polynomials with coefficients modulo a zero-dimensional triangular set. Our algorithm generalizes previous work for computing GCDs over algebraic number fields. The main difficulty is when a zero divisor is encountered modulo a prime number. We give two ways of handling this: Hensel lifting, and fault tolerant rational reconstructi...

Let 6n be the symmetric group of n elements and g(n) = max (order of c). We give here some effective bounds for g(n) and P(g(n)) (greatest prime divisor of g(n)). Theoretical proofs are in "Evaluation asymptotique de l'ordre maximum d'un élément du groupe symétrique" (Acta Arith., v. 50, 1988, pp. 221-242). The tools used here are techniques of superior highly composite numbers of Ramanujan and...