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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:
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