General Harmonic Numbers in terms of Riemann Zeta and Hurwitz Zeta Functions
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Theorem
- $\harm r x = \map \zeta r - \map \zeta {r, x + 1}$
where:
- $\harm r x$ denotes the general harmonic number of order $r$ evaluated at $x$
- $\map \zeta r$ is the Riemann zeta function
- $\map \zeta {r, x}$ is the Hurwitz zeta function
- $r$ and $x$ are complex numbers with $\map \Re r > 1$ and $x \notin \Z_{<0}$
Proof
\(\ds \harm r x\) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \paren {\frac 1 {k^r} - \frac 1 {\paren {k + x}^r} }\) | Definition of General Harmonic Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \frac 1 {k^r} - \sum_{k \mathop = 1}^{\infty} \frac 1 {\paren {k + x}^r}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \frac 1 {k^r} - \sum_{k \mathop = 0}^{\infty} \frac 1 {\paren {k + \paren {x + 1} }^r}\) | reindexing the sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \zeta r - \map \zeta {r, x + 1}\) | Definition of Hurwitz Zeta Function and Definition of Riemann Zeta Function |
$\blacksquare$
Sources
- 1985: Bruce C. Berndt: Ramanujan's Notebooks: Part I: Chapter $7$. Sums of Powers, Bernoulli Numbers, and the Gamma Function