Difference between revisions of "Euler constant"
From Encyclopedia of Mathematics
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The number $\gamma$ defined by | The number $\gamma$ defined by | ||
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$$ 1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln(n+1)$$ | $$ 1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln(n+1)$$ | ||
| − | is monotone increasing and bounded from above. The number-theoretic nature of the Euler constant has not been studied; it is not even known ( | + | is monotone increasing and bounded from above. |
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| + | The number $\gamma$ is also known as the ''Euler-Mascheroni'' constant, after L. Euler (1707–1783) and L. Mascheroni (1750–1800). | ||
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| + | The number-theoretic nature of the Euler constant has not been studied; it is not even known (2022) whether it is a rational number or not. | ||
In fact, a relation | In fact, a relation | ||
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holds, cf. {{Cite|HaWr|Chapter 22.5}}. | holds, cf. {{Cite|HaWr|Chapter 22.5}}. | ||
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====Comments==== | ====Comments==== | ||
Indeed, one also has | Indeed, one also has | ||
| − | + | $$ | |
| − | + | \gamma = -\psi(1) = -\Gamma'(1) = \sum_{k=1}^\infty \left[{\frac{1}{k} - \log\left(1 - \frac{1}{k} \right)}\right] = - \int_0^\infty e^{-t}\log t\,dt | |
| − | + | $$ | |
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and | and | ||
| + | $$ | ||
| + | \gamma = \sum_{k=1}^\infty \frac{z}{k(k+z)} - \psi(z+1) = 2 \sum_{k=1}^n \frac{1}{2k-1} - 2\log 2 - \psi(n+1/2) | ||
| + | $$ | ||
| + | for $z \in \mathbb{C} \setminus \mathbb{Z}^{-}$, $\mathbb{Z}^{-} = \mathbb{Z}_0^{-} \setminus \{0\}$, $n \in \mathbb{N}_0 = \mathbb{N} \cup \{0\}$, and where an empty sum is interpreted, as usual, to be zero. In terms of the [[Riemann zeta function]] $\zeta(s)$, Euler's classical results state: | ||
| + | $$ | ||
| + | \gamma = \sum_{k=2}^\infty (-1)^k \frac{\zeta(k)}{k} = \log 2 - \sum_{k=1}^\infty \frac{\zeta(2k+1)}{2k+1} 2^{-2k}\ . | ||
| + | $$ | ||
| − | + | ====References==== | |
| − | + | * {{Ref|HaWr}} G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 {{MR|0568909}} {{ZBL|0423.10001}} | |
| − | + | * {{Ref|Fi}} Steven R. Finch, "Mathematical constants" , Encyclopedia of mathematics and its applications '''94''', Cambridge University Press (2003) {{ISBN|0-521-81805-2}} {{ZBL|1054.00001}} | |
| − | + | * {{Ref|a1}} A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, "Higher transcendental functions" , '''I''' , McGraw-Hill (1953) | |
| − | + | * {{Ref|a2}} L. Lewin, "Polylogarithms and associated functions" , Elsevier (1981) | |
| − | + | * {{Ref|a3}} H.M. Srivastava, J. Choi, "Series associated with the zeta and related functions" , Kluwer Acad. Publ. (2001) | |
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Latest revision as of 11:50, 23 November 2023
The number $\gamma$ defined by
$$ \gamma=\lim_{n\to \infty}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n\right)\approx 0.57721566490\ldots,$$
considered by L. Euler (1740). Its existence follows from the fact that the sequence
$$ 1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln(n+1)$$
is monotone increasing and bounded from above.
The number $\gamma$ is also known as the Euler-Mascheroni constant, after L. Euler (1707–1783) and L. Mascheroni (1750–1800).
The number-theoretic nature of the Euler constant has not been studied; it is not even known (2022) whether it is a rational number or not.
In fact, a relation
$$ \sum_{n\leq x}\,\frac{1}{n}-\ln x=\gamma+O\left(\frac{1}{x}\right)$$
holds, cf. [HaWr, Chapter 22.5].
Comments
Indeed, one also has $$ \gamma = -\psi(1) = -\Gamma'(1) = \sum_{k=1}^\infty \left[{\frac{1}{k} - \log\left(1 - \frac{1}{k} \right)}\right] = - \int_0^\infty e^{-t}\log t\,dt $$ and $$ \gamma = \sum_{k=1}^\infty \frac{z}{k(k+z)} - \psi(z+1) = 2 \sum_{k=1}^n \frac{1}{2k-1} - 2\log 2 - \psi(n+1/2) $$ for $z \in \mathbb{C} \setminus \mathbb{Z}^{-}$, $\mathbb{Z}^{-} = \mathbb{Z}_0^{-} \setminus \{0\}$, $n \in \mathbb{N}_0 = \mathbb{N} \cup \{0\}$, and where an empty sum is interpreted, as usual, to be zero. In terms of the Riemann zeta function $\zeta(s)$, Euler's classical results state: $$ \gamma = \sum_{k=2}^\infty (-1)^k \frac{\zeta(k)}{k} = \log 2 - \sum_{k=1}^\infty \frac{\zeta(2k+1)}{2k+1} 2^{-2k}\ . $$
References
- [HaWr] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 MR0568909 Zbl 0423.10001
- [Fi] Steven R. Finch, "Mathematical constants" , Encyclopedia of mathematics and its applications 94, Cambridge University Press (2003) ISBN 0-521-81805-2 Zbl 1054.00001
- [a1] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, "Higher transcendental functions" , I , McGraw-Hill (1953)
- [a2] L. Lewin, "Polylogarithms and associated functions" , Elsevier (1981)
- [a3] H.M. Srivastava, J. Choi, "Series associated with the zeta and related functions" , Kluwer Acad. Publ. (2001)
How to Cite This Entry:
Euler constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_constant&oldid=35955
Euler constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_constant&oldid=35955
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article