ISSN 0032-9460, Problems of Information Transmission, 2014, Vol. 50, No. 2, pp. 186–202.
Pleiades Publishing, Inc., 2014.
Original Russian Text
E.A. Karatsuba, 2014, published in Problemy Peredachi Informatsii, 2014, Vol. 50, No. 2, pp. 77–95.
On One Method for Fast Approximation of Zeta
Constants by Rational Fractions
E. A. Karatsuba
Dorodnitsyn Computing Center of the Russian Academy of Sciences, Moscow, Russia
Received November 12, 2013; in ﬁnal form, March 21, 2014
Abstract—We present a new method for deriving both known and new fast approximations
of zeta constants ζ(n), n ≥ 2, n is an integer, by rational fractions.
Zeta constants are values of the Riemann zeta function ζ(s) for integer arguments s = n, n ≥ 2.
The Dirichlet series
converges too slowly and does not yield a fast approximation to ζ(n). At the same time, fast ap-
proximations of zeta constants are needed for both fast evaluation of the constants and analyzing
the arithmetic structure of these values of the Riemann zeta function. In the last case, of spe-
cial interest are fast approximations of zeta constants by rational fractions. Such approximations
for particular zeta constants and methods for deriving them were considered in numerous papers
Presently, for fast evaluation (on algorithms for fast evaluation of transcendental functions,
see [19–30]) of zeta constants, three representations are used.
First, for fast evaluation of the zeta constants ζ(2n), n ≥ 1, with accuracy 2
one can construct
a fast algorithm (see ) with complexity O(M(N )log
N), where M(N)isthecomplexityof
multiplication of two N-digit numbers, based on using the well-known formula (see, e.g., )
are Bernoulli coeﬃcients deﬁned by recurrent relations
Second, for any n, n ≥ 2, one can apply the fast evaluation algorithm for ζ(n) constructed
in [24, 25] based on the FEE method (see [19–30]), which also has complexity O(M(N)log
this case, the approximation
(n − 1)!
(i − 1)! G