An efficient algorithm for the computation of Bernoulli numbers

Abstract

This article gives a direct formula for the computation of B(n) using the asymptotic formula $$B (n) \approx 2 {\frac {n!}{{\pi}^{n}{2}^{n}}}$$ where n is even and $n >> 1$. This is simply based on the fact that $\zeta (n)$ is very near 1 when n is large and since $B (n) = 2 {\frac {\zeta (n) n!}{{\pi}^{n}{2}^{n}}}$ exactly. The formula chosen for the Zeta function is the one with prime numbers from the well-known Euler product for $\zeta (n)$. This algorithm is far better than the recurrence formula for the Bernoulli numbers even if each B(n) is computed individually. The author could compute $B (750,000)$ in a few hours. The current record of computation is now (as of Feb. 2007) $B (5,000,000)$ a number of (the numerator) of 27332507 decimal digits is also based on that idea.