One of the odd zeta values from \(\zeta(5)\) to \(\zeta(25)\) is irrational. By elementary means. (English) Zbl 1445.11063

Zudilin, Wadim

SIGMA, Symmetry Integrability Geom. Methods Appl. 14, Paper 028, 8 p. (2018).

Let \(\zeta(k)=\sum_{n=1}^\infty\frac 1{n^k}\) be the zeta function. Then the author proves that at least one of the numbers \(\zeta(5), \zeta(7), \dots ,\zeta(25)\) is an irrational number. The proof is not simple but uses only elementary tools like prime number theorem, Stirling’s formula \(n!=\sqrt{2\pi n} (\frac ne)^n\) and so on.

Reviewer: Jaroslav Hančl (Ostrava)

MSC:

11J72	Irrationality; linear independence over a field
11M06	\(\zeta (s)\) and \(L(s, \chi)\)
33C20	Generalized hypergeometric series, \({}_pF_q\)

Keywords:

irrationality; zeta value; hypergeometric series; rational function

References:

[1]	Ball, Keith and Rivoal, Tanguy, Irrationalit\'e d’une infinit\'e de valeurs de la fonction z\^eta aux entiers impairs, Inventiones Mathematicae, 146, 1, 193-207, (2001) · Zbl 1058.11051
[2]	de Bruijn, N. G., Asymptotic methods in analysis, Bibliotheca Mathematica, 4, xii+200, (1958), North-Holland Publishing Co., Amsterdam, P. Noordhoff Ltd., Groningen, Interscience Publishers Inc., New York · Zbl 0082.04202
[3]	Fischler, St\'ephane, Irrationalit\'e de valeurs de z\^eta (d’apr\`es {A}p\'ery, {R}ivoal, {\( \dots \)}), Ast\'erisque, 294, no. 294, 27-62, (2004) · Zbl 1101.11024
[4]	Fischler, St\'ephane and Sprang, Johannes and Zudilin, Wadim, Many odd zeta values are irrational · Zbl 1398.11109
[5]	Hanson, Denis, On the product of the primes, Canadian Mathematical Bulletin. Bulletin Canadien de Math\'ematiques, 15, 33-37, (1972) · Zbl 0231.10008
[6]	Krattenthaler, Christian and Zudilin, Wadim, Hypergeometry inspired by irrationality questions · Zbl 1450.11072
[7]	Rivoal, Tanguy, Irrationalit\'e d’au moins un des neuf nombres {\( \zeta(5),\zeta(7),\dots,\zeta(21)\)}, Acta Arithmetica, 103, 2, 157-167, (2002) · Zbl 1015.11033
[8]	Rivoal, Tanguy and Zudilin, Wadim, A note on odd zeta values · Zbl 1470.11203
[9]	Sprang, Johannes, Infinitely many odd zeta values are irrational. By elementary means · Zbl 1398.11109
[10]	Zudilin, Wadim, Arithmetic of linear forms involving odd zeta values, Journal de Th\'eorie des Nombres de Bordeaux, 16, 1, 251-291, (2004) · Zbl 1156.11327