# Hypergeometry inspired by irrationality questions

@article{Krattenthaler2018HypergeometryIB,
title={Hypergeometry inspired by irrationality questions},
author={C. Krattenthaler and W. Zudilin},
journal={arXiv: Number Theory},
year={2018}
}
• Published 2018
• Mathematics
• arXiv: Number Theory
We report new hypergeometric constructions of rational approximations to Catalan's constant, $\log2$, and $\pi^2$, their connection with already known ones, and underlying `permutation group' structures. Our principal arithmetic achievement is a new partial irrationality result for the values of Riemann's zeta function at odd integers.
6 Citations
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Many values of the Riemann zeta function at odd integers are irrational
• Mathematics
• 2018
Abstract In this note, we announce the following result: at least 2 ( 1 − e ) log ⁡ s log ⁡ log ⁡ s values of the Riemann zeta function at odd integers between 3 and s are irrational, where e is anyExpand
A note on odd zeta values
• Mathematics
• 2018
Using a new construction of rational linear forms in odd zeta values and the saddle point method, we prove the existence of at least two irrational numbers amongst the 33 odd zeta values ζ(5), ζ(7),.Expand
Arithmetic of Catalan’s constant and its relatives
We prove that at least one of the six numbers $$\beta (2i)$$β(2i) for $$i=1,\ldots ,6$$i=1,…,6 is irrational. Here $$\beta (s)=\sum _{k=0}^{\infty }(-1)^k(2k+1)^{-s}$$β(s)=∑k=0∞(-1)k(2k+1)-s denotesExpand
Many odd zeta values are irrational
• Mathematics
• Compositio Mathematica
• 2019
Building upon ideas of the second and third authors, we prove that at least $2^{(1-\unicode[STIX]{x1D700})(\log s)/(\text{log}\log s)}$ values of the Riemann zeta function at odd integers between 3Expand
One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means
Available proofs of result of the type "at least one of the odd zeta values $\zeta(5),\zeta(7),\dots,\zeta(s)$ is irrational" make use of the saddle-point method or of linear independence criteria,Expand

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Abstract In this note, we announce the following result: at least 2 ( 1 − e ) log ⁡ s log ⁡ log ⁡ s values of the Riemann zeta function at odd integers between 3 and s are irrational, where e is anyExpand
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One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means
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