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# Seminario / Sprang

On Friday November 9th, 11.30-13.00, Aula dottorato

Johannes SPRANG (Regensburg) will talk about

Title: Some new irrationality results for ($p$-adic) zeta values

Abstract: It has already been known to Euler that the values of the Riemann zeta function at positive even integers are non-zero rational multiples of powers of $\pi$. Much less is known about the values at positive odd integers.
The irrationality of $\zeta(3)$ has been proven by Ap\'ery and a celebrated theorem of Rivoal and Ball shows that the dimension of the $\mathbb{Q}$-vector space spanned by $1,\zeta(3),\zeta(5),...,\zeta(s)$ for an odd positive integer $s$ is at least $C\log(s)$ for an absolute and explicit constant $C$. In particular, at least $C\log(s)$ among these numbers are irrational. In our recent work with Fischler and Zudilin, we improve this lower bound on the number of irrational zeta values to $2^{(1-\epsilon)\frac{\log(s)}{\log\log(s)}}$. The main ingredient is the construction of sufficiently many linear independent families of linear forms in zeta values with related coefficients. If time permits, we will explain how a related construction can be used to prove a $p$-adic variant of the theorem of Rivoal and Ball.

07 novembre 2018