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
for tending to infinity, where and are coprime integers.
Theorem 4.1. For every there is an integer with , , and
Proof. We will need some basic facts on the arithmetical functions and . Let
where is restricted on primes. Moreover, let for positive integers . Then,
where (4.5) follows from [5, Theorem 420] and the prime number theorem. By [5, Theorem 118] (von Staudt's theorem) we know how to obtain the prime divisors of the denominators of Bernoulli numbers : The denominators of are squarefree, and they are divisible exactly by those primes with . Hence,
Next, let ( are the subscripts of in Theorem 2.4). First, we consider the following terms from the series transformation in :
with
For every there is a rational defined by
where , , , and
Similarly, we define rationals by
where , and . We have
Therefore, using the conclusion (4.6) from von Staudt's theorem, we get
Note that , since every integer with divides at least one integer with .
From (4.10) and (4.13) we conclude on
Hence we have from (4.4) and (4.5) that
The theorem is proved.
Remark 4.2. On the one side we have shown that and . On the other side, every prime dividing satisfies and therefore divides . Conversely, all primes with divide , but not . That means: is much bigger than , but is formed by powers of small primes, whereas is divisible by many big primes.
5. Simplification of the Transformed Series
Let
such that
In Theorem 2.4 the sequence is transformed. In view of a simplified process we now investigate the transformation of the series . Therefore we have to estimate the contribution of to the series transformation in Theorem 2.4. For this purpose, we define
A major step in estimating is to expressthe sums on the right-hand side by integrals.
Lemma 5.1. For positive integers and one has
Proof. For integers and a real number with the identity
holds, which we apply with and to substitute the fraction . Introducing the new variable , we then get
The sum inside the brackets of the integrand can be expressed by using the equation
in which we put and . This gives the identity stated in the lemma.
The following result deals with the case , in which we express the finite sum by a double integral on a rational function.
Corollary. For every positive integer one has
Proof. Set in Lemma 5.1, and note that
Hence,
Let be any positive integer. Then we have the following decomposition of a rational function, in which is considered as variable and as parameter:
We additionally assume that . Then, differentiating this identity -times with respect to , the polynomial in on the right-hand side vanishes identically:
Therefore, we get from (5.10) by iterated integrations by parts:
The corollary is proved by noting that
6. Estimating
In this section we estimate defined in (5.3). Substituting for into the integral in Lemma 5.1 and applying iterated integration by parts, we get
Set
where and are kept fixed. We have . For an integer we use Cauchy's formula
to estimate . Let denote the circle in the complex plane centered around 0 with radius . With and defined above, Cauchy's formula yields the identity
For the complex arithm function occurring in (6.4) we cut the complex plane along the negative real axis and exclude the origin by a small circle. All arguments of a complex number are taken from the interval . Therefore, using , we get
Hence,
Thus, it follows from (6.4) that
From we conclude on
Since is a strictly increasing function, we get
For , this upper bound also holds for . Finally, we note that . Altogether, we conclude from (6.7) on
It follows that the Taylor series expansion of ,
converges at least for . Then,
and the estimate given by (6.10) implies for that
Combining (6.13) with the result from (6.1), we get for
We estimate the binomial coefficient by Stirling's formula (3.12). For this purpose we additionally assume that :
We now assume and substitute the above inequality into (6.14):
For all integers and we have
Thus we have proven the following result.
Lemma 6.1. For all integers with and one has
Next, we need an upper bound for the Bernoulli numbers (cf. [9, 23.1.15] ):
Let and . Using this and Lemma 6.1, we estimate in (5.3):
Now, let
with the numbers introduced in the proof of Theorem 4.1. By definition of and we then have , and therefore we can estimate the series transformation of by applying the results from Theorem 2.4 and (6.20). Again, let and .
By similar arguments we get the same bound when . For it can easily be seen that
Thus, we finally have proven the following theorem.
Theorem 6.2. Let
where are positive integers. Let be an integer. Then, there is a positive constant depending at most on and such that
7. Concluding Remarks
It seems that in Theorem 6.2 a smaller bound holds.
Conjecture. Let be positive integers. Let be an integer. Then there is a positive constant depending at most on and such that for all integers one has
A proof of this conjecture would be implied by suitable bounds for the integral stated in Lemma 5.1. For such a bound follows from the double integral given in Corollary 5.2:
where the double integral in the last but one line equals to .
Note that the rational functions
take their maximum values and inside the unit square at and , respectively. Finally, we compare the bound for the series transformation given by Theorem 2.4 with the bound proven for Theorem 6.2. In Theorem 2.4 the bound is
whereas we have in Theorem 6.2 that
For fixed and sufficiently large it is clear on the one hand that , but on the other hand we have
Conversely, for tending to infinity, one gets
with implicit constants depending at most on . For the denominators of the transformed series in Theorem 2.4 we have the bound from Theorem 4.1, and a similar inequality holds for the denominators of the transformed series in Theorem 6.2.