Monotone and fast computation of Euler’s constant
- José A Adell†^{1}Email author and
- Alberto Lekuona†^{1}
https://doi.org/10.1186/s13660-017-1507-8
© The Author(s) 2017
Received: 3 May 2017
Accepted: 30 August 2017
Published: 15 September 2017
Abstract
We construct sequences of finite sums \((\tilde{l}_{n})_{n\geq 0}\) and \((\tilde{u}_{n})_{n\geq 0}\) converging increasingly and decreasingly, respectively, to the Euler-Mascheroni constant γ at the geometric rate 1/2. Such sequences are easy to compute and satisfy complete monotonicity-type properties. As a consequence, we obtain an infinite product representation for \(2^{\gamma }\) converging in a monotone and fast way at the same time. We use a probabilistic approach based on a differentiation formula for the gamma process.
Keywords
MSC
1 Introduction
As far as we know, two main types of computations of the Euler constant have been developed. The first one emphasizes the monotonicity of the corresponding convergent sequences, but the rates of convergence are relatively slow (polynomial rates). The second one emphasizes the speed of convergence (geometric rates), but looses the monotonicity in the approximation.
In this paper, we gather both points of view by providing approximating sequences that converge at the geometric rate 1/2 and satisfy complete monotonicity-type properties (monotonicity, convexity, etc.). In addition, such approximating sequences are easy to compute.
We point out that some authors have introduced probabilistic tools to deal with different topics of analytic number theory. For instance, Sun [12] described Stirling series in terms of products of uniformly distributed random variables, Srivastava and Vignat [13] have given representations of the Bernoulli, Euler, and Gegenbauer polynomials in terms of moments of appropriate random variables, and Ta [14] has recently introduced a nice probabilistic approach to study Appell polynomials by connecting them to moments of random variables. Finally, fast computations of the Stieltjes constants using differentiation formulas for linear operators represented by stochastic processes can be found in [15] and the references therein.
2 Main results
Theorem 2.1
The approximating sequences to γ given in (2) and (3) are simpler to compute than those in Theorem 2.1. However, the sequences \((\tilde{l}_{n})_{n\geq 0}\) and \((\tilde{u}_{n})_{n\geq 0}\) in this theorem converge to γ in a much faster way and enjoy properties such as monotonicity, convexity, and so on. On the other hand, formulas (4), (5), (6), and (7) provide fast computations of γ at the price of loosing the monotonicity of the corresponding approximating sequences. Certainly, formula (7) computes γ in a faster way than that in (13). However, the sequences \((\tilde{l}_{n})_{n\geq 0}\) and \((\tilde{u}_{n})_{n\geq 0}\) in Theorem 2.1 are easier to compute than the main term in (7).
Corollary 2.2
3 Auxiliary results
This formula can be applied to the problem at hand as follows.
Lemma 3.1
Proof
Lemma 3.2
Proof
Thanks to Lemma 3.2, the complete monotonicity-type properties of \((l_{n})_{n\geq 0}\) and \((u_{n})_{n\geq 0}\) are easily derived from the analogous properties satisfied by the sequences of functions \((L_{n}(x))_{n\geq 0}\) and \((U_{n}(x))_{n\geq 0}\).
Lemma 3.3
Proof
4 The proofs
Proof of Theorem 2.1
Proof of Corollary 2.2
Notes
Declarations
Acknowledgements
The authors are partially supported by Research Projects DGA (E-64), MTM2015-67006-P, and by FEDER funds.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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