ISSN 0032-9460, Problems of Information Transmission, 2015, Vol. 51, No. 4, pp. 378–390.
Pleiades Publishing, Inc., 2015.
Original Russian Text
E.A. Karatsuba, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 4, pp. 78–91.
On One Method for Constructing a Family
of Approximations of Zeta Constants
by Rational Fractions
E. A. Karatsuba
Dorodnitsyn Computing Center of the Federal Research Center
“Computer Science and Control” of the Russian Academy of Sciences, Moscow, Russia
e-mail: firstname.lastname@example.org, email@example.com
Received July 21, 2015; in ﬁnal form, September 7, 2015
Abstract—We present a new method for fast approximation of zeta constants, i.e., values
ζ(n) of the Riemann zeta function, n ≥ 2, n is an integer, by rational fractions. The method
makes it possible to fast approximate zeta constants and certain combinations of successive
values of zeta constants by rational fractions. By choosing values of coeﬃcients involved in the
combinations, one can control the convergence rate of the approximations and the computation
complexity for the zeta constants.
In [1, 2] a new method was constructed for fast approximations of zeta constants, i.e., val-
ues ζ(n) of the Riemann zeta function, n is an integer, n ≥ 2, by rational fractions, based on
Kummer-type transformations. In , by an example of some zeta constants it was shown that
using multifactorials allows to somewhat increase the convergence rate of such approximations.
Another way to control the approximations of zeta constants arises based on the Hermite–Beukers
approach (see [3, 4]). In recent years, this approach with the use of shifted Legendre polynomials
and also generalizations of this approach have been widely used in proving irrationality and analyz-
ing the arithmetic structure of some number-theoretic constants, including certain zeta constants
Note that none of the two above-mentioned methods can yet be applied for fast evaluation of
any zeta constants. Presently, the only method making it possible to fast evaluate the Riemann
zeta function at any integer point is the method from [21–24] with the use of the FEE process
(see [21–31]). The computation complexity of this method is close to optimal (see [21–24]).
Nevertheless, new formulas for zeta constants given in the present paper can be useful for both
evaluating them and analyzing the arithmetic structure of these numbers. The scheme of presenting
the results is aimed at obtaining approximations suitable for evaluating the zeta constants, whereas
the analysis of the arithmetic nature of these numbers is not made. Therefore, the presentation
order diﬀers from what was used in [3–20]. Some of the formulas derived below are generalizations
of previously obtained ones. However, using the method presented in this paper, completely new
results can also be obtained.
Supported in part by the Russian Foundation for Basic Research, project no. 13-01-00657.