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Faster computation of Bernoulli numbers. (English) Zbl 0755.11006

The author presents an algorithm, based on the classical formula

B 2k =(-1) k+1 2(2k!)ζ(2k)(2π) -2k ,

to compute the 2kth Bernoulli number B 2k , defined by X/(e X -1)= n1 B n X n /n!. The space requirement of this algorithm is 𝒪(nloglogn) bits and it involves 𝒪(n 2 log 2 nloglogn) bit operations. The algorithm can also be efficiently generalized to compute all Bernoulli numbers up to some point. This slightly improves on the previous algorithms of S. Chowla and P. Hartung [Acta Arith. 22, 113-115 (1972; Zbl 0244.10008)] and of D. E. Knuth and T. J. Buckholtz [Math. Comput. 21, 663-688 (1967; Zbl 0178.044)].

11B68Bernoulli and Euler numbers and polynomials
11Y16Algorithms; complexity (number theory)
68Q25Analysis of algorithms and problem complexity