Fachhochschule für die Wirtschaft Hannover, Freundallee 15, 30173 Hannover, Germany
The author continues to study series transformations for
the Euler-Mascheroni constant . Here, we discuss in detail recently published
results of A. I. Aptekarev and T. Rivoal who found rational approximations to and () defined by linear recurrence formulae. The main purpose of
this paper is to adapt the concept of linear series transformations with integral
coefficients such that rationals are given by explicit formulae which approximate and . It is shown that for every and every integer there are infinitely many rationals for such that and with for tending to infinity.
1. Introduction
Let
It is well known that the sequence converges to Euler's constant , where
Nothing is known on the algebraic background of such mathematical constants like Euler's constant . So we are interested in better diophantine approximations of these numbers, particularly in rational approximations.
In 1995 the author [1] introduced a linear transformation for the series with integer coefficients which improves the rate of convergence. Let be an additional positive integer parameter.
Proposition 1.1 (see [1]). For any integers and one has
Particularly, by choosing , one gets the following result.
Corollary 1.2. For any integer one has
Some authors have generalized the result of Proposition 1.1 under various aspects. At first one cites a result due to Rivoal [2].
Proposition (see [2]). For tending to infinity, one has
Kh. Hessami Pilehrood and T. Hessami Pilehrood have found some approximation formulas for the logarithms of some infinite products including Euler's constant . These results are obtained by using Euler-type integrals, hypergeometric series, and the Laplace method [3].
Proposition ([3]). For tending to infinity the following asymptotic formula holds:
Recently the author has found series transformations involving three parameters , and , [4]. In Propositions 1.5 and 1.6 certain integral representations of the (discrete) series transformations are given, which exhibit important (analytical) tools to estimate the error terms of the transformations.
Proposition 1.5 (see [4]). Let , and be integers. Additionally one assumes that
Then one has
Proposition (see [4]). Let , and be integers. Additionally one assumes that
Then one has
with
Setting
one gets an explicit upper bound from Proposition 1.6
Corollary 1.7. For integers , , one has
where is some constant depending only on . For one gets
For an application of Corollary 1.7 let the integers and be defined by
denotes the von Mangoldt function. By [5, Theorem 434] one has
Then, for , there is some integer such that
Multiplying (1.14) by , we deduce the following corollary.
Corollary. There is an integer such that one has for all integers that
2. Results on Rational Approximations to
In 2007, Aptekarev and his collaborators [6] found rational approximations to , which are based on a linear third-order recurrence. For the sake of brevity, let .
Proposition (see [6]). Let and be two solutions of the linear recurrence
with , , and , , . Then, one has , , and
with two positive constants . It seems interesting to replace the fraction by
and to estimate the remainder in terms of .
Corollary 2.2. Let . Then there are two positive constants , such that for all sufficiently large integers one has
Recently, Rivoal [7] presented a related approach to the theory of rational approximations to Euler's constant , and, more generally, to rational approximations for values of derivatives of the Gamma function. He studied simultaneous Padé approximants to Euler's functions, from which he constructed a third-order recurrence formula that can be applied to construct a sequence in that converges subexponentially to for any complex number . Here, is defined by its principal branch. We cite a corollary from [7].
Proposition 2.3 (see [7]). (i) The recurrence
provides two sequences of rational numbers and with , , and , , such that converges to .(ii) The recurrence
provides two sequences of rational numbers and with , , and , , such that converges to .
The goal of this paper is to construct rational approximations to without using recurrences by a new application of series transformations. The transformed sequences of rationals are constructed as simple as possible, only with few concessions to the rate of convergence (see Theorems 2.4 and 6.2 below).
In the following we denote by the Bernoulli numbers, that is, , , , and so on (In Sections 3–6 the Bernoulli numbers cannot be confused with the integers from Corollary 2.2.) In this paper we will prove the following result.
Theorem 2.4. Let , , and be positive integers, and
Then,
where is some positive constant depending only on .
3. Proof of Theorem 2.4
Lemma 3.1. One has for positive integers and
Proof. Applying the well known inequality , we get
This proves the lemma.
takes its maximum value for with
which leads to a better bound than in Lemma 3.1. But we are satisfied with Lemma 3.1. A main tool in proving Theorem 2.4 is Euler's summation formula in the form
where is a suitable chosen parameter, and the remainder is defined by a periodic Bernoulli polynomial , namely
with
Applying the summation formula to the function , we get (see [8, equation ( 5)] )
It follows that
We prove Theorem 2.4 for . The case is treated similarly. So we have again by the above summation formula that
First, we estimate the integral on the right-hand side of (3.8). We have
since . Next, we assume that . Hence , and therefore we estimate the integral on the right-hand side in (3.9) by
In the sequel we put . Moreover, in the above formula we now replace by with . In order to estimate we use Stirling's formula
Then, it follows that
and similarly we have
By using the definition of in Theorem 2.4, the formula (1.1) for , and the identities (3.8), (3.9), it follows that
where is specified to and to . Moreover, we know from [4, Lemma 2] that
By setting , the above formula for the series transformation of simplifies to
where , and . Here, we have used the results from Corollary 1.7, (3.13), and (3.14). The sum
vanishes, since for every real number we have
where on the right-hand side for an integer with one term in the numerator equals to zero.
The inequality
holds for all integers . Now, using Lemma 3.1, we estimate the right-hand side in (3.17) for and as follows:
The last but one estimate holds for all integers , , and is a suitable positive real constant depending on . This completes the proof of Theorem 2.4.
4. On the Denominators of
In this section we will investigate the size of the denominators of our series transformations