Euler constant

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)