Sum of General Harmonic Numbers in terms of Riemann Zeta Function/Corollary

Corollary to Sum of General Harmonic Numbers in terms of Riemann Zeta Function

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

where:

$\harm r x$ and $\harm r {-\dfrac k n}$ denotes the general harmonic number of order $r$ evaluated at $x$ and $\paren {\dfrac {-k} n}$, respectively.

$\map \zeta r$ is the Completed Riemann zeta function

$r$ and $x$ are complex numbers

$n \in \Z_{>0}$.

Proof

\(\ds \harm r x - \dfrac 1 {n^r} \sum_{k \mathop = 0}^{n - 1} \harm r {x - \dfrac k n}\)

\(=\)

\(\ds \paren {1 - n^{1 - r} } \map \zeta r\)

Sum of General Harmonic Numbers in terms of Riemann Zeta Function

\(\ds \leadsto \ \ \)

\(\ds \harm r 0 - \dfrac 1 {n^r} \sum_{k \mathop = 0}^{n - 1} \harm r {0 - \dfrac k n}\)

\(=\)

\(\ds \paren {1 - n^{1 - r} } \map \zeta r\)

Setting $x:= 0$

\(\ds \leadsto \ \ \)

\(\ds -\dfrac 1 {n^r} \sum_{k \mathop = 1}^{n - 1} \harm r {-\dfrac k n}\)

\(=\)

\(\ds \paren {1 - n^{1 - r} } \map \zeta r\)

Harmonic Number Zero: $H_0 = 0$

\(\ds \leadsto \ \ \)

\(\ds \paren {-n^r} \times - \dfrac 1 {n^r} \sum_{k \mathop = 1}^{n - 1} \harm r {-\dfrac k n}\)

\(=\)

\(\ds \paren {-n^r} \times \paren {1 - n^{1 - r} } \map \zeta r\)

multiplying both sides by $-n^r$

\(\ds \leadsto \ \ \)

\(\ds \sum_{k \mathop = 1}^{n - 1} \harm r {-\dfrac k n}\)

\(=\)

\(\ds \paren {n - n^r} \map \zeta r\)

$\blacksquare$

Sources

• 1985: Bruce C. Berndt: Ramanujan's Notebooks: Part I: Chapter $7$. Sums of Powers, Bernoulli Numbers, and the Gamma Function