Numerical calculation of the Riemann zeta function at odd-integer arguments: a direct formula method

In this article, we introduce a recurrence formula which only involves two adjacent values of the Riemann zeta function at integer arguments. Based on the formula, an algorithm to evaluate $$\zeta $$ζ-values (i.e., the values of Riemann zeta function) at odd integers from the two nearest $$\zeta $$ζ-values at even integers is posed and proved. The behavior of the error bound is $$O(10^{-n})$$O(10-n) approximately where $$n$$n is the argument. Our method is especially powerful for the calculation of Riemann zeta function at large argument, while for smaller ones, it can also reach spectacular accuracies such as more than ten decimal places.