# An accurate approximation of zeta-generalized-Euler-constant functions

Open Mathematics (2010)

• Volume: 8, Issue: 3, page 488-499
• ISSN: 2391-5455

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## Abstract

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Zeta-generalized-Euler-constant functions, $\gamma \left(s\right):=\sum _{k=1}^{\infty }\left(\frac{1}{{k}^{s}}-{\int }_{k}^{k+1}\frac{dx}{{x}^{s}}\right)$ and $\stackrel{˜}{\gamma }\left(s\right):=\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}\left(\frac{1}{{k}^{s}}-{\int }_{k}^{k+1}\frac{dx}{{x}^{s}}\right)$ defined on the closed interval [0, ∞), where γ(1) is the Euler-Mascheroni constant and $\stackrel{˜}{\gamma }$ (1) = ln $\frac{4}{\pi }$ , are studied and estimated with high accuracy.

## How to cite

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Vito Lampret. "An accurate approximation of zeta-generalized-Euler-constant functions." Open Mathematics 8.3 (2010): 488-499. <http://eudml.org/doc/269603>.

@article{VitoLampret2010,
abstract = {Zeta-generalized-Euler-constant functions, $\gamma \left( s \right): = \sum \limits \_\{k = 1\}^\infty \{\left( \{\frac\{1\}\{\{k^s \}\} - \int \_k^\{k + 1\} \{\frac\{\{dx\}\}\{\{x^s \}\}\} \} \right)\}$ and $\tilde\{\gamma \}\left( s \right): = \sum \limits \_\{k = 1\}^\infty \{\left( \{ - 1\} \right)^\{k + 1\} \left( \{\frac\{1\}\{\{k^s \}\} - \int \_k^\{k + 1\} \{\frac\{\{dx\}\}\{\{x^s \}\}\} \} \right)\}$ defined on the closed interval [0, ∞), where γ(1) is the Euler-Mascheroni constant and $\tilde\{\gamma \}$ (1) = ln $\frac\{4\}\{\pi \}$ , are studied and estimated with high accuracy.},
author = {Vito Lampret},
journal = {Open Mathematics},
keywords = {Alternating; Convergence acceleration; Estimate; Generalized-Euler-constant-function; Inequality; Series; Zeta; alternating; convergence acceleration; estimate; generalized-Euler-constant-function; inequality; series; zeta},
language = {eng},
number = {3},
pages = {488-499},
title = {An accurate approximation of zeta-generalized-Euler-constant functions},
url = {http://eudml.org/doc/269603},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Vito Lampret
TI - An accurate approximation of zeta-generalized-Euler-constant functions
JO - Open Mathematics
PY - 2010
VL - 8
IS - 3
SP - 488
EP - 499
AB - Zeta-generalized-Euler-constant functions, $\gamma \left( s \right): = \sum \limits _{k = 1}^\infty {\left( {\frac{1}{{k^s }} - \int _k^{k + 1} {\frac{{dx}}{{x^s }}} } \right)}$ and $\tilde{\gamma }\left( s \right): = \sum \limits _{k = 1}^\infty {\left( { - 1} \right)^{k + 1} \left( {\frac{1}{{k^s }} - \int _k^{k + 1} {\frac{{dx}}{{x^s }}} } \right)}$ defined on the closed interval [0, ∞), where γ(1) is the Euler-Mascheroni constant and $\tilde{\gamma }$ (1) = ln $\frac{4}{\pi }$ , are studied and estimated with high accuracy.
LA - eng
KW - Alternating; Convergence acceleration; Estimate; Generalized-Euler-constant-function; Inequality; Series; Zeta; alternating; convergence acceleration; estimate; generalized-Euler-constant-function; inequality; series; zeta
UR - http://eudml.org/doc/269603
ER -

## References

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1. [1] Abramowitz M., Stegun I. A., Handbook of Mathematical Functions, 9th ed., Dover Publications, N.Y., 1972 Zbl0543.33001
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4. [4] Lampret V., Approximating real Pochhammer products: A comparison with powers, Cent. Eur. J. Math., 2009, 7, 493–505 http://dx.doi.org/10.2478/s11533-009-0036-1 Zbl1179.41034
5. [5] Sîntămărian A., A generalization of Euler’s constant, Numer. Algorithms, 2007, 46, 141–151 http://dx.doi.org/10.1007/s11075-007-9132-0 Zbl1130.11075
6. [6] Sîntămărian A., Some inequalities regarding a generalization of Euler’s constant, J. Inequal. Pure Appl. Math., 2008, 9(2), 46 Zbl1195.11168
7. [7] Sondow J., Double integrals for Euler’s constant and ln $\frac{4}{\pi }$ and an analog of Hadjicosta’s formula, Amer. Math. Monthly, 2005, 112, 61–65 http://dx.doi.org/10.2307/30037385 Zbl1138.11356
8. [8] Sondow J., Hadjicostas P., The generalized-Euler-constant function γ(z) and a generalization of Somos’s quadratic recurrence constant, J. Math. Anal. Appl., 2007, 332, 292–314 http://dx.doi.org/10.1016/j.jmaa.2006.09.081 Zbl1113.11017
9. [9] Wolfram S., Mathematica, Version 6.0, Wolfram Research, Inc., 1988–2008

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