Sum of General Harmonic Numbers in terms of Riemann Zeta Function

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Theorem

$\ds \harm r x - \dfrac 1 {n^r} \sum_{k \mathop = 0}^{n - 1} \harm r {x - \dfrac k n} = \paren {1 - n^{1 - r} } \map \zeta r$

where:

$\harm r x$ and $\harm r {x - \dfrac k n}$ denotes the general harmonic number of order $r$ evaluated at $x$ and $\paren {\dfrac {x - k} n}$, respectively.
$\map \zeta r$ is the Completed Riemann zeta function
$r$ and $x$ are complex numbers with $\paren {\dfrac {x - k} n} \notin \Z_{<0}$
$n \in \Z_{>0}$


Corollary

$\ds \sum_{k \mathop = 1}^{n - 1} \harm r {-\dfrac k n} = \paren {n - n^r} \map \zeta r$


Proof

Lemma

$\ds \sum_{j \mathop = 1}^\infty \frac 1 {\paren {j + x}^r} = \sum_{k \mathop = 0}^{n - 1} \sum_{j \mathop = 1}^\infty \frac 1 {\paren {n j - k + x }^r}$

$\Box$


\(\ds \harm r x - \dfrac 1 {n^r} \sum_{k \mathop = 0}^{n - 1} \harm r {x - \dfrac k n}\) \(=\) \(\ds \sum_{j \mathop = 1}^\infty \paren {\frac 1 {j^r} - \frac 1 {\paren {j + x}^r} } - \dfrac 1 {n^r} \sum_{k \mathop = 0}^{n - 1} \sum_{j \mathop = 1}^\infty \paren {\frac 1 {j^r} - \frac 1 {\paren {j + \paren {\dfrac {x - k} n} }^r} }\) Definition of General Harmonic Numbers
\(\ds \) \(=\) \(\ds \sum_{j \mathop = 1}^\infty \frac 1 {j^r} - \sum_{j \mathop = 1}^{\infty} \frac 1 {\paren {j + x}^r} - \dfrac 1 {n^r} \sum_{k \mathop = 0}^{n - 1} \sum_{j \mathop = 1}^\infty \frac 1 {j^r} + \dfrac 1 {n^r} \sum_{k \mathop = 0}^{n - 1} \sum_{j \mathop = 1}^\infty \frac 1 {\paren {j + \paren {\dfrac {x - k} n} }^r}\) Sum of Absolutely Convergent Series
\(\ds \) \(=\) \(\ds \sum_{j \mathop = 1}^\infty \frac 1 {j^r} - \sum_{j \mathop = 1}^\infty \frac 1 {\paren {j + x}^r} - \dfrac n {n^r} \sum_{j \mathop = 1}^\infty \frac 1 {j^r} + \sum_{k \mathop = 0}^{n - 1} \sum_{j \mathop = 1}^\infty \frac 1 {\paren {n j - k + x }^r}\) simplifying
\(\ds \) \(=\) \(\ds \map \zeta r - \sum_{j \mathop = 1}^\infty \frac 1 {\paren {j + x}^r} - \dfrac n {n^r} \map \zeta r + \sum_{k \mathop = 0}^{n - 1} \sum_{j \mathop = 1}^\infty \frac 1 {\paren {n j - k + x }^r}\) Definition of Riemann Zeta Function
\(\ds \) \(=\) \(\ds \map \zeta r - \dfrac n {n^r} \map \zeta r\) Lemma
\(\ds \) \(=\) \(\ds \paren {1 - n^{1 - r} } \map \zeta r\) simplifyiing

$\blacksquare$


Sources

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