Faster Computation of Bernoulli Numbers

Abstract Using the relationship between the Riemann zeta function and Bernoulli numbers, we develop an algorithm for the fast computation of Bernoulli numbers of high index. We present one algorithm to compute all Bernoulli numbers up to and including B 2 n , and a modified algorithm to compute B 2 n directly. We show that in both algorithms, all computations can be done using at most [2 n log 2 n ] bits and that the complexity of the first algorithm is O ( n 2 log n ) multiplications of numbers at most O ( n log n ) bits while that of the second is O ( n ) multiplications of numbers of at most O ( n log n ) bits.