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Recently I have been playing around with series of the form

$$\sum_{k=1}^{\infty}\frac{k^{s}}{e^{kz}-1} = \sum_{k=1}^{\infty}\sigma_{s}(k)e^{-kz}$$

for $s \in \mathbb{Z}$ and where $\sigma_s(k)$ is the sum of divisors function of order $s$. These series have generated quite a bit of interest over the years, due in large part to some beautiful modular identities of Ramanujan. The most famous example being

$$\alpha^{-n}\left(\frac{1}{2}\zeta(2n+1)+\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\alpha k}-1}\right) = \\ (-\beta)^n\left(\frac{1}{2}\zeta(2n+1)+\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\beta k}-1}\right) - 2^{2n}\sum_{k=0}^{n+1}(-1)^k\frac{B_{2k}}{(2k)!}\frac{B_{2n+2-2k}}{(2n+2-2k)!}\alpha^{n+1-k}\beta^k$$

where $\alpha,\beta > 0, \alpha\beta=\pi^2$ and $B_k$ are the Bernoulli numbers and $\zeta(k)$ is the Riemann zeta function. As far as I know there aren't any similar relations or closed forms when $s \in 2\mathbb{Z}$. In my investigations I was able to find some approximation formula for general $s > 0$ but which unfortunately perform poorer and poorer as $s \rightarrow \infty$.

For instance, at $s=2$ we have

$$\sum_{k=1}^{\infty}\frac{k^2z}{e^{kz}-1} \approx \frac{2\zeta(3)}{z^2} - \frac{1}{2}-\frac{z}{24} -\sum_{j=0}^{N}B^{(2)}_{j+2}B_{j}\frac{z^{j}}{(j+2)!}$$

where $B^{(k)}_n$ are the Norlund polynomials.

I was excited to find this, but then unfortunately realized that since the sum on the right hand side diverges as $N \rightarrow \infty$ we can only achieve a finite number of accurate digits as the RHS approaches the left from below then surpasses it, growing without bound.

For instance letting $N=37$ we have

$$\sum_{k=1}^{\infty}\frac{k^2}{e^{k}-1} \approx 2 \zeta (3)-\frac{707928034947324016593079681811720894660110227517}{8567110474102926210628918330759216889856000000000}$$

with the right hand side being correct to the 14-th decimal place. This is about the best we can do with the above formula.

I am curious as to whether someone would be able to provide a better approximation. I am not very familiar with sort of thing... so maybe there is a standard way of achieving approximations like the one above?

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