Web Page of the Ottawa-Carleton Number Theory Seminar

Year 2016-2017

Title: On prime numbers with as many ones as zeros in their binary expansion.

Abstract: Drmota, Mauduit and Rivat showed in 2009 that the set E of prime numbers with as many ones as zeros in their binary expansion is infinite (they actually gave an asymptotic formula for the number of elements of E in [1, N] for N≥ 2). We prove that for every irrational number β, the sequence ( βp)_p∈E is uniformly distributed modulo 1.

This is a joint work with Christian Mauduit and Joël Rivat (Université d'Aix-Marseille).

Abstract: For q square-free, we call an integer s a square modulo q if and only if s is a square modulo p for all primes p dividing q. In this talk we look the variance of the distribution of squares modulo a composite number. We also look at inverse questions for the large sieve in distribution aspect and make some improvements regarding distribution of tuples of reduced residues.

Abstract: This talk is about approximation to successive powers of a real number by rational numbers with the same denominator. The quality of the approximation is compared to the size of the denominator. We also study several variations of the above problem, in terms of classical exponents of Diophantine approximation. We discuss open questions like Wirsing's problem concerning approximation to real numbers by algebraic numbers of degree at most n, for a given integer n.

Résumé : Une variété abélienne définie sur un corps de nombres et admettant des multiplications complexes (CM) a potentiellement bonne réduction partout. Lorsqu'une courbe de genre non nul a bonne réduction en une place finie, sa variété jacobienne aussi. La réciproque est toutefois fausse dès le genre 2. Dans un article en commun avec Philipp Habegger, nous montrons le résultat suivant : Soit F un corps quadratique réel. Il n'y a qu'un nombre fini de courbes C de genre 2 definies sur une clôture algébrique de Q (à isomorphisme près) dont la jacobienne Jac(C) est CM par un ordre maximal d'une extension K cyclique, quartique, contenant F et qui ont potentiellement bonne réduction partout. Une telle courbe aura donc presque toujours au moins une place de mauvaise réduction stable, alors que sa jacobienne a bonne réduction partout.

Abstract : I will speak on a recent joint work with Jean-Marc Deshouillers, Sanoli Gun and Florian Luca. Here is a sample result. Let τ(.) be the classical Ramanujan τ-function defined by
q∏_n>0 (1-qⁿ)²⁴ = ∑_n>0 τ(n) qⁿ .
The classical work of Rankin implies that both inequalities |τ(n)|<|τ(n+1)| and |τ(n)|>|τ(n+1)| hold for infinitely many n. We generalize this for longer segments of consecutive values of τ.
Let k be a positive integer such that τ(n) is not 0 for n≤ k/2. (This is known to be true for all k < 10²³, and, conjecturally, holds for all k.) Let s be a permutation of the set {1,...,k}. Then there exist infinitely many positive integers n such that |τ(n+s(1))|<|τ(n+s(2))|<...<|τ(n+s(k))|.

Title: Continued fraction of certain Mahler functions and applications to Diophantine approximation.

Abstract We consider a class of Mahler functions g(x) which can be written as an infinite product g(x) = 1/x * \prod_{t=0}^\infty P(x^{-d^t}), where d is a positive integer and P(x) is a polynomial of degree less than d. In this talk we show that the continued fraction of g(x), written as a Laurent series, can be computed by a recurrent formula. Then we will use this fact to establish several approximational properties of Mahler numbers g(b) for integer b>1 and some functions g(x). In particular we will compute their irrationality exponent in some cases and make non-trivial estimates on it in the other cases. Also, if time permits, we will show that the Thue-Morse number is not badly approximable.

Abstract Definition of Mahler numbers induces a natural construction of infinite sequences of good rational approximations. A bit surprisingly, in some cases these sequences of approximations imply that the corresponding Mahler number is badly approximable, whilst in the others it allows to provide a non-trivial lower bound for its irrationality measure. We are going to discuss such estimates. This will be in continuation with the talk by Dzmitry Badziahin.

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