Séminaire TNGA : trois exposés de Carlo Sanna, Antonella Perucca et Florian Luca

Exposé de Carlo Sanna (Politecnico di Torino)

Titre : Fibonacci partitions, automata, and generalized spectral radiis

Résumé : For each positive integer $n$, let $r_F(n)$ be the number of ways to write $n$ as a sum of distinct Fibonacci numbers, where the order of the summands does not matter. What is the average of $r_F(n)$ ? More generally, what is the $k$th moment of $r_F(n)$ ? Chown, Jones, and Slattery answered these questions for $k=1,2$ via counting arguments and inequalities. In turns out that, thanks to ideas of Berstel and Shallit, these questions can be answered by employing automata theory and a result on the generalized spectral radius.

Exposé de Antonella Perucca (Université du Luxembourg)

Titre : Almost 100 years of Artin’s conjecture on primitive roots

Résumé : If $a$ is an integer and $p$ a prime number, we say that $a$ is a
primitive root modulo $p$ if the powers of $a$ are representants for all
the non-zero residue classes modulo $p$. In 1927, Artin conjectured
that, provided that $a$ is neither a square nor $-1$, the set of primes
$p$ such that $a$ is a primitive root modulo $p$ has a positive natural
density. In 1967, Hooley was able to prove Artin’s conjecture assuming GRH. Assuming GRH, with Järviniemi and Sgobba we have obtained very general results on Artin-type problems, and with Shparlinski we have discovered uniform bounds related to the Artin density.

Exposé de Florian Luca (Stellenbosch University et Max Planck Institute for Software Systems, Saarbrücken)

Titre : On large zeros of linear recurrence sequences

Résumé : The Skolem Problem asks to determine whether a given integer linear recurrence sequence (LRS) has a zero term. In my talk, I introduce a notion of « large » zeros of (non-degenerate) linear recurrence sequences, i.e., zeros occurring at an index larger than a sixth-fold exponential of the size of the data defining the given LRS. We establish two main results. First, we show that large zeros are very sparse: the set of positive integers that can possibly arise as large zeros of some LRS has null density. Second, we define an infinite set of prime numbers, termed « good », having density one amongst all prime numbers, with the following property: for any large zero of a given LRS, there is an interval around the large zero together with an upper bound on the number of good primes possibly present in that interval. The bound in question is much lower than one would expect if good primes were distributed similarly as ordinary prime numbers, as per the Cramér model in number theory. We therefore conjecture that large zeros do not exist, which would entail decidability of the Skolem Problem. This is joint work with J. Ouaknine (MPI-SWS) and J. Worrell (Oxford).