# 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} }

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

Hypergeometric rational approximations to ζ(4)

- Mathematics
- Proceedings of the Edinburgh Mathematical Society
- 2020

Abstract We give a new hypergeometric construction of rational approximations to ζ(4), which absorbs the earlier one from 2003 based on Bailey's 9F8 hypergeometric integrals. With the novel… Expand

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 any… Expand

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

- Mathematics
- 2018

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 denotes… Expand

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 3… Expand

One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means

- Mathematics
- 2018

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 any… Expand

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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

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