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

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.

### MSC:

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