One of $\zeta(7),\zeta(9),\dots,\zeta(21)$ is irrational
We prove that at least one of the eight numbers $\zeta(7),\zeta(9),\dots,\zeta(21)$ is irrational. The previous best result of this kind not involving $\zeta(5)$, a consequence of a theorem of Lai and Zhou, was that one of $\zeta(7),\dots,\zeta(35)$ is irrational. We use Zudilin's hypergeometric construction: sums of fourth derivatives of rational functions with quintuple zeros give linear forms in $1,\zeta(7),\dots,\zeta(21)$ with no $\zeta(3)$ or $\zeta(5)$ term. The improvement comes from spreading the poles over eighteen blocks with nearby offsets, a sharper $p$-adic bound on the…
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