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Irrationality of odd zeta values and other treasures

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Wadim Zudilin
Radboud University Nijmegen
Wed, 2018-09-05 09:25 - 10:10
MPIM Lecture Hall

Proving that a given number, like the value of Riemann's zeta function $\zeta(s)$ at an integer $s\ge2$, is irrational may be challenging.
It is known that all even zeta values $\zeta(2k)$, where $k=1,2,\dots$, are irrational and, thanks to Ap\'ery (1978), that $\zeta(3)$ is irrational.
Nothing of this sort can be said about the (ir)rationality of odd zeta values $\zeta(2k+1)$, where $k=1,2,\dots$, though a theorem of Rivoal and Ball (2000) demonstrates that infinitely many of these numbers happen to be irrational.
More precisely, the theorem tells that of the numbers $\zeta(5),\zeta(7),\dots,\zeta(2k+1)$ at least $c\log k$ are irrational, for some absolute constant $c>0$.
In our recent work with Fischler and Sprang we produce a new construction of simultaneous $\mathbb Q$-linear forms in odd zeta values and replace the Ball--Rivoal bound $c\log k$ with $\exp\{(c\log k)/(\log\log k)\}$, something "much more like a power of $k$ than a power of $\log k$'' according to Hardy and Wright.
I shall outline the principal ingredients of this construction and, if time allows, explain its extension to related classes of numbers, for example, to Catalan's constant and its relatives.

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