On Rational Approximations to Euler ’ s Constant γ and to γ log a / b

The author continues to study series transformations for the Euler-Mascheroni constant γ . Here, we discuss in detail recently published results of A. I. Aptekarev and T. Rivoal who found rational approximations to γ and γ log q q ∈ Q>0 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 γ and γ log q. It is shown that for every q ∈ Q>0 and every integer d ≥ 42 there are infinitely many rationals am/bm for m 1, 2, . . . such that |γ log q − am/bm| 1 − 1/d / d − 1 4 m and bm | Zm with logZm ∼ 12d2m2 form tending to infinity.


Introduction
Let It is well known that the sequence s n n≥1 converges to Euler's constant γ 0, 577 . .., where Nothing is known on the algebraic background of such mathematical constants like Euler's constant γ.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 s n n≥1 with integer coefficients which improves the rate of convergence.Let τ be an additional positive integer parameter.
Particularly, by choosing τ n ≥ 2, one gets the following result.
Corollary 1.2.For any integer n ≥ 2one has 1.4 Some authors have generalized the result of Proposition 1.1 under various aspects.At first one cites a result due to Rivoal 2 .

Proposition 1.3 see 2 . For n tending to infinity, one has
1.5 Kh. Hessami Pilehrood and T. Hessami Pilehrood have found some approximation formulas for the logarithms of some infinite products including Euler's constant γ.These results are obtained by using Euler-type integrals, hypergeometric series, and the Laplace method 3 .Proposition 1. 4 3 .For n tending to infinity the following asymptotic formula holds: 1.6 Then one has 1.8 Proposition 1.6 see 4 .Let n ≥ 1, τ 1 ≥ 1 and τ 2 ≥ 1 be integers.Additionally one assumes that 1.9 Then one has 1 − ut n 1 du dt, 1.10

1.11
Setting 12 one gets an explicit upper bound from Proposition 1.6

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For an application of Corollary 1.7 let the integers B m and A m be defined by

Corollary 1.8.
There is an integer m 0 such that one has for all integers m ≥ m 0 that

Results on Rational Approximations to γ
In 2007, Aptekarev and his collaborators 6 found rational approximations to γ, which are based on a linear third-order recurrence.For the sake of brevity, let D n l.c.m. 1, 2, . . ., n .
Proposition 2.1 see 6 .Let p n n≥0 and q n n≥0 be two solutions of the linear recurrence with p 0 0, p 1 2, p 2 31/2 and q 0 1, q 1 3, q 2 25.Then, one has q n ∈ Z, D n p n ∈ Z, and with two positive constants c 0 , c 1 .
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2.4
Recently, Rivoal 7 presented a related approach to the theory of rational approximations to Euler's constant γ, 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 Q z that converges subexponentially to log z γ for any complex number z ∈ C \ −∞, 0 .Here, log is defined by its principal branch.We cite a corollary from 7 .provides two sequences of rational numbers p n n≥0 and q n n≥0 with p 0 −1, p 1 4, p 2 77/4 and q 0 1, q 1 7, q 2 65/2 such that p n /q n n≥0 converges to γ.
(ii) The recurrence provides two sequences of rational numbers p n n≥0 and q n n≥0 with p 0 −1, p 1 11, p 2 71 and q 0 0, q 1 8, q 2 56 such that p n /q n n≥0 converges to log 2 γ.

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The goal of this paper is to construct rational approximations to γ log a/b 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 B 2n the Bernoulli numbers, that is, B 2 1/6, B 4 −1/30, B 6 1/42, and so on In Sections 3-6 the Bernoulli numbers cannot be confused with the integers B n from Corollary 2.2.In this paper we will prove the following result.

2.7
Then, where c 4 is some positive constant depending only on d.

Proof of Theorem 2.4
Lemma 3.1.One has for positive integers d and m Proof.Applying the well known inequality This proves the lemma.
g k takes its maximum value for k k 0 with which leads to a better bound than 16 dm 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 r ∈ N is a suitable chosen parameter, and the remainder R r is defined by a periodic Bernoulli polynomial 3.5 with Applying the summation formula to the function f x 1/x, we get see 8, equation 5 It follows that We prove Theorem 2.4 for a ≥ b.The case a < b is treated similarly.So we have again by the above summation formula that

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First, we estimate the integral on the right-hand side of 3.8 .We have Next, we assume that n ≥ a. Hence bn, an ⊆ n, n 2 , and therefore we estimate the integral on the right-hand side in 3.9 by

3.11
In the sequel we put r dm.Moreover, in the above formula we now replace n by dm k with 0 ≤ k ≤ dm.In order to estimate 2r !we use Stirling's formula Then, it follows that

3.13
International Journal of Mathematics and Mathematical Sciences 9 and similarly we have

3.14
By using the definition of S n in Theorem 2.4, the formula 1.1 for s n , and the identities 3.8 , 3.9 , it follows that where r is specified to r dm and n to n dm k.Moreover, we know from 4, Lemma 2 that

3.16
By setting n dm k, the above formula for the series transformation of S dm k simplifies to 3.17 vanishes, since for every real number x > −dm we have where on the right-hand side for an integer x with −m ≤ x ≤ d − 1 m − 1 one term in the numerator equals to zero.The inequality holds for all integers d ≥ 42.Now, using Lemma 3.1, we estimate the right-hand side in 3.17 for dm ≥ a and d ≥ 42 as follows:

3.21
The last but one estimate holds for all integers m ≥ 2, d ≥ 42, and c 4 is a suitable positive real constant depending on d.This completes the proof of Theorem 2.4.

On the Denominators of S n
In this section we will investigate the size of the denominators b m of our series transformations Proof.We will need some basic facts on the arithmetical functions ϑ x and ψ x .Let where p is restricted on primes.Moreover, let D n : l.c.m 1, 2, . . ., n for positive integers n.Then, Next, let max{a, b} ≤ dm ≤ n ≤ 2dm n k dm are the subscripts of S k dm in Theorem 2.4 .First, we consider the following terms from the series transformation in S m : International Journal of Mathematics and Mathematical Sciences with a < b .

4.8
For every m ≥ 1 there is a rational x m /y m defined by

4.11
where u m ∈ Z, v m ∈ N and u m , v m 1.We have

4.15
The theorem is proved.
Remark 4.2.On the one side we have shown that log Y m ∼ 4d 2 m 2 and log V m ∼ 8d 2 m 2 .On the side, every prime p dividing V m satisfies p ≤ max{a, b, dm, 2dm 1, 2dm} 2dm 1 and therefore p divides Y m D 4d 2 m 2 .Conversely, all primes p with 2dm 1 < p < 4d 2 m 2 divide Y m , but not V m .That means: V m is much bigger than Y m , but V m is formed by powers of small primes, whereas Y m is divisible by many big primes.

Simplification of the Transformed Series
In Theorem 2.4 the sequence S n is transformed.In view of a simplified process we now investigate the transformation of the series S n − R n .Therefore we have to estimate the contribution of R k dm to the series transformation in Theorem 2.4.For this purpose, we define

5.3
A major step in estimating E m is to express the sums on the right-hand side by integrals.

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Lemma 5.1.For positive integers d, j and m one has Proof.For integers k, r and a real number ρ with k ρ > 0 the identity holds, which we apply with r 2j and ρ dm to substitute the fraction 1/ dm k 2j .Introducing the new variable u : e −t , we then get

5.6
The sum inside the brackets of the integrand can be expressed by using the equation in which we put n dm and τ d − 1 m − 1.This gives the identity stated in the lemma.
The following result deals with the case j 1, in which we express the finite sum by a double integral on a rational function.

Corollary 5.2. For every positive integer m one has
Proof.Set j 1 in Lemma 5.1, and note that International Journal of Mathematics and Mathematical Sciences 15 Hence,

5.10
Let s be any positive integer.Then we have the following decomposition of a rational function, in which u is considered as variable and w as parameter: We additionally assume that s − 1 < dm.Then, differentiating this identity dm-times with respect to u, the polynomial in u on the right-hand side vanishes identically: 5.12 Therefore, we get from 5.10 by iterated integrations by parts:

5.13
The corollary is proved by noting that 5.14

Estimating E m
In this section we estimate E m defined in 5.3 .Substituting 1 − u for u into the integral in Lemma 5.1 and applying iterated integration by parts, we get where m and j are kept fixed.We have f 0 0. For an integer k > 0 we use Cauchy's formula to estimate |f k 0 |.Let C denote the circle in the complex plane centered around 0 with radius R : 1 − 1/2k.With a 0 and f z defined above, Cauchy's formula yields the identity For the complex logarithm 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 φ of a complex number z / ∈ −∞, 0 are taken from the interval −π, π .Therefore, using 1−Re iφ 1−R cos φ− iR sin φ, we get

6.6
Thus, it follows from 6.4 that

6.8
Since arctan is a strictly increasing function, we get arctan 6.9 For 0 < R < 1, this upper bound also holds for φ 0. Finally, we note that R k 1 − 1/2k k ≥ 1/2.Altogether, we conclude from 6.7 on

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It follows that the Taylor series expansion of f u , 6.11 converges at least for −1 < u < 1.Then, 12 and the estimate given by 6.10 implies for 0 < u < 1 that

6.13
Combining 6.13 with the result from 6.1 , we get for m > 1

6.14
We estimate the binomial coefficient by Stirling's formula 3.12 .For this purpose we additionally assume that m ≥ 2d − 1:

6.15
We now assume m ≥ 2d − 1 and substitute the above inequality into 6.14 :

International Journal of Mathematics and Mathematical Sciences 19
For all integers m ≥ 1 and d ≥ 1 we have 6.17 Thus we have proven the following result.

6.18
Next, we need an upper bound for the Bernoulli numbers B 2j cf. 9, 23.1.15:

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Now, let

6.22
By similar arguments we get the same bound when b > a.For d ≥ 3 it can easily be seen that

2 International
Journal of Mathematics and Mathematical Sciences Proposition 1.1 see 1 .For any integers n ≥ 1 and τ ≥ 2 one has

10 0 − 1
International Journal of Mathematics and Mathematical Scienceswhere dm ≥ a, m ≥ 2, and d ≥ 3. Here, we have used the results from Corollary 1.7, 3.13 , and 3.14 .The sum dm k dm k g k dm k 3.18

− 1 Theorem 4 . 1 .
dm k g k S k dm , 4.1 for m tending to infinity, where a m ∈ Z and b m ∈ N are coprime integers.For every m ≥ 1 there is an integer Z m with Z m > 0, b m |Z m , and

5 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 B 2k : The denominators of B 2k are squarefree, and they are divisible exactly by those primes p with p − 1 | 2k.Hence,

4 . 9 where
x m ∈ Z, y m ∈ N, x m , y m 1, andy m | Y m : D 4d 2 m 2 ,dm ≥ max{a, b} .4.10Similarly, we define rationals u m /v m by

Lemma 6 . 1 .
For all integers d, m with d ≥ 3 and m ≥ 2d − 1 one has dm

2d d d − 1 d− 1 2d − 1 0 − 1 kConjecture 7 . 1 . 0 − 1 < c 6 • 1 − 1 1 A 1 − 1 0 1 0 1 − u 2 1 − w u 2 1 − 1 − u w 3 1 − u u 2 w 1 − 1 − u w d− 2 × 1 − 1 1 − 1 − 1 − u u 2 w 1 − 1 − u w , 1 − u d 1 − w u 2d−1 w d− 1 1 − 1 − u w d , 7 . 3 take
where a, b are positive integers.Let d ≥ 42 be an integer.Then, there is a positive constant c 5 depending at most on a, b and d such that dm k 2d − 1 m k − 1 dm dm k T k dm − γ − log a It seems that in Theorem 6.2 a smaller bound holds.Let a, b be positive integers.Let d ≥ 2 be an integer.Then there is a positive constant c 6 depending at most on a, b and d such that for all integers m ≥ 1 one has dm k dm k 2d − 1 m k − 1 dm dm k T k dm − γ − log a b proof of this conjecture would be implied by suitable bounds for the integral stated in Lemma 5.1.For j 1 such a bound follows from the double integral given in Corollary 5.2: u dm 1 − w m u 2d−1 m−1 w d−1 m−1 1 − 1 − u w dm 1 du dw u d 1 − w u 2d−1 w d−integral in the last but one line equals to 1/24.Note that the rational functions their maximum values 4 2−d and 1 − 1/d d / d −1 4 d inside the unit square 0, 1 × 0, 1 at u, w 1/2, 1 and u, w 1/2, 2d − 2 / 2d − 1 , 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 T 1 d, m : c 4 • 1 − 1/d d d − 1 4 d m , d ≥ 42, m ≥ 1 , 7.4

5
It seems interesting to replace the fraction p n /q n by Corollary 2.2.Let 0 < ε < 1.Then there are two positive constants c 2 , c 3 , such that for all sufficiently large integers n one has introduced in the proof of Theorem 4.1.By definition of R n and S n we then have T n S n − R n , and therefore we can estimate the series transformation of T n by applying the results from Theorem 2.4 and 6.20 .Again, let m ≥ max{2d−1, a/2} and d ≥ 42.