For questions regarding harmonic numbers, which are partial sums of the harmonic series. The $N$-th harmonic number is the sum of reciprocals of the first $N$ natural numbers.
The $n$-th harmonic number $H_n$ is defined by
$$H_n = \sum\limits_{k = 1}^n \frac{1}{k}$$
The harmonic numbers are important in various fields of number theory, and have been studied since antiquity. The harmonic numbers are known to grow slowly, tending to infinity at roughly the same rate as the natural logarithm: $$H_n\propto \gamma +\ln(n)$$ Where $\gamma$ is the euler-mascheroni-constant
The definition of harmonic numbers can also be extended to the complex plane: $$H_z=\gamma+\psi(z+1)$$ Where $\psi(z)$ is the digamma-function
Generalized harmonic numbers are also defined: $$H_{n}^{(m)}:=\sum_{k=1}^{n}\frac{1}{k^m}$$
The definition of generalized harmonic numbers can also be extended to the complex plane:
$$H_{z}^{(s)}:=\zeta(s)-\zeta(s,z+1)$$ Where $\zeta(s)$ is the riemann-zeta and $\zeta(s,z)$ is the Hurwitz zeta function.
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