Appears in collection : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire

This talk will report on a work with S. Bettin (University of Genova) in which we obtained exact modularity relations for the q-Pochhammer symbol, which is a finite version of the Dedekind eta function. We will overview some of their useful aspects and applications, in particular to the value distribution of a certain knot invariants, the Kashaev invariants, constructed with q-Pochhammer symbols.