One of β(4), β(6),..., β(36) is irrational
Let $\beta(s)=\sum_{m\ge 0}(-1)^m(2m+1)^{-s}$ be Dirichlet's beta function. We prove that at least one of the seventeen numbers $\beta(4),\beta(6),\dots,\beta(36)$ is irrational. All previous explicit results of this kind that we know of include $\beta(2)$, Catalan's constant; our statement does not involve $\beta(2)$ at all. The proof uses the linear forms $L_n=\sum_{t\ge0}(-1)^tR_n''(t)$ attached to a very-well-poised rational function $R_n$ with triple integer zeros and half-integer poles. The second derivative, a $\beta$-analogue of a device of Rivoal, removes $\beta(2)$, and a symmetry…
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