La 4ième petite journée GANDA

IMB le 27 mai 2021

Résumés des exposés

Florian LUCA (Johannesburg et Saarbrücken)
Some results on the Ramanujan tau-function
Let \(\tau(n)\) be the Ramanujan \(\tau\)-function of \(n\). In 2009, Alkan, Ford and Zaharescu proved that for every real number \(\beta\) there is a constant \(C_{\beta}\) such that the Diophantine inequality \[ \left|\frac{\tau(n)}{n^{11/2}}-\beta\right|<\frac{C_{\beta}}{\log n} \] has infinitely many integer solutions \(n\ge 2\). In this talk, I will survey some of my recent results concerning distributional properties of the Ramanujan \(\tau\)-function which were inspired by the above result. I will also mention some results on prime factors of \(\tau(n)\) and its iterates. For example, assuming the Lehmer conjecture that \(\tau(n)\ne 0\) for all \(n\), if \(n\) is even and \(k\ge 1\), then \(\tau^{(k)}(n)\) is divisible by a prime \(p\ge 3^{k-1}+1\). Given a fixed finite set of odd primes \(S=\{p_1,\ldots,p_\ell\}\) we give a bound on the number of solutions \(n\) to the equation \(\tau(n)=\pm p_1^{a_1}\cdots p_\ell^{a_\ell}\) for integers \(a_1,\ldots,a_\ell\) and in case \(S:=\{3,5,7\}\), we show that there is no such \(n>1\).

Fabien PAZUKI (Copenhague et Bordeaux)
Propriété de Northcott pour les valeurs spéciales de fonctions L
On discute d'un travail en cours avec Riccardo Pengo, où la question posée est la suivante. Considérons une famille \(F\) d'objets \(X\) munis d'une fonction \(L(X,s)\). On fixe un entier naturel \(n\), et on suppose que la suite des valeurs \(L(X,n)\), pour \(X\) variant dans \(F\), est bornée. Peut-on conclure que la famille \(F\) est finie ?

Yuri BILU (Bordeaux)
Binary polynomial power sums vanishing at roots of unity
I will speak on a recent work with Florian Luca, where we show that, apart some obvious exceptions, the polynomials of the form \(a(x)f(x)^n + b(x)g(x)^n\) cannot vanish at roots of unity of order exceeding some effective constant. The qualitative result follows from the Bombieri-Masser-Zannier-Maurin work on unlikely intersections, but our contribution is making this effective (in fact, totally explicit), which, to our knowledge, cannot be deduced from any available version of the BMZ or similar theorem.

Avec de soutien de l'IMB, de l'IRN CNRS « GANDA » et du projet ANR JINVARIANT

           

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