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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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