The Hurwitz zeta function $\zeta(s, a)$ admits a well-known (divergent) asymptotic expansion in powers of $1/a$ involving the Bernoulli numbers. Using Wilson orthogonal polynomials, we determine an effective bound for the error made when this asymptotic series is replaced by nearly diagonal Pade approximants. By specialization, we obtain new fast converging sequences of rational approximations to the values of the Riemann zeta function at every integers $\ge 2$. The latter can be viewed, in a certain sense, as analogues of Apery's celebrated sequences of rational approximations to $\zeta(2)$ and $\zeta(3)$.

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Hurwitz zeta 函数的余数 Padé 近似的对角收敛

Hurwitz zeta 函数 $\zeta(s, a)$ 承认涉及伯努利数的 $1/a$ 幂的众所周知的（发散的）渐近展开。使用威尔逊正交多项式，我们确定了当这个渐近级数被接近对角的 Pade 近似代替时所产生的误差的有效界限。通过专门化，我们在每个整数 $\ge 2$ 处获得了黎曼 zeta 函数值的有理近似值的新快速收敛序列。在某种意义上，后者可以看作是 Apery 著名的 $\zeta(2)$ 和 $\zeta(3)$ 有理近似序列的类似物。