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Difference between revisions of "Euler constant"

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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 (2012) whether it is a rational number or not.
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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
 
In fact, a relation
  
$$ \sum_{n\leq x}\frac{1}{n}-\ln x=\gamma+O\left(\frac{1}{x}\right)$$
+
$$ \sum_{n\leq x}\,\frac{1}{n}-\ln x=\gamma+O\left(\frac{1}{x}\right)$$
  
 
holds, cf. {{Cite|HaWr|Chapter 22.5}}.
 
holds, cf. {{Cite|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====
 
====References====
{|
+
 
|-
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* {{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}}
|valign="top"|{{Ref|HaWr}}||valign="top"|  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapts. 5; 7; 8
+
* {{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)

Revision as of 06:19, 10 April 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=25612
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article