How do I calculate $\sum_{n\geq1}\frac{1}{n^4+1}$?

How do I calculate the following sum $$\sum_{n\geq1}\frac{1}{n^4+1}$$

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Hint Consider the function $f(z) = \dfrac{\pi \cot \pi z}{z^4+1}$. I assume you have seen other examples of computing series with residue calculus (since you marked the question complex-analysis).

A few more details. Let $C(z) = \pi\cot \pi z$. Then $C$ is holomorphic everywhere except at the integers, and $\newcommand{\Res}{\operatorname{Res}}\Res(C;z=n) = 1$ for every $n \in \mathbb{Z}$. The function $f$ has additional simple poles at points where $z^4+1=0$.

Let $\gamma_N$ be the positively oriented square with corners at $\pm(N+\frac12) \pm(N+\frac12)i$. A tedious computation will show that $|C(z)|$ is bounded on $\gamma_N$ by a constant, not depending on $N$. (It you haven’t seen this, try to work out the details yourself. Check a textbook if you can’t get it.)

The residue theorem gives

$$\int_{\gamma_N} f(z)\,dz = \sum_{k=-N}^N \Res(f;z=k) + \sum_{\alpha^4=-1} \Res(f;z=\alpha).$$

Letting $N\to\infty$, we end up with (since $|f|\to0$ quickly enough at $\infty$):

$$0 = \sum_{k=-\infty}^\infty \frac{1}{k^4+1} + \sum_{\alpha^4=-1} \Res(f;z=\alpha).$$

What remains is to compute the four extra residues, and manipulate the doubly infinite series a little, but since it’s homework, I’m not going to finish things off for you.

First, we write the sum in the following form

$$ \sum_{n=0}^{\infty}\frac{1}{n^4+1} = \frac{i}{2}\sum_{n=0}^{\infty}\frac{1}{n^2+i} -\frac{i}{2} \sum_{n=0}^{\infty}\frac{1}{n^2-i}\quad i=\sqrt{-1}, $$

Then, we use the result

$$\sum_{n=0}^{\infty}\frac{1}{n^2+a^2} =\frac{\pi}{a} \frac{e^{2a\pi}}{e^{2a\pi}-1} \,. $$

$$\sum_{n=1}^\infty \frac{1}{n^4+1}=\zeta(4)-\zeta(8)+\zeta(12)-\zeta(16)+\zeta(20)…$$
$$\frac{\pi t}{e^{2\pi t}-1}-\frac{1}{2}+\frac{\pi t}{2} = \zeta(2) t^2 – \zeta(4) t^4 + \zeta(6) t^6 – \zeta(8)t^8+\zeta(10)t^{10}\cdots.$$
$$\frac{\pi ti}{e^{2\pi t i}-1}-\frac{1}{2}+\frac{\pi t i}{2} = -\zeta(2) t^2 – \zeta(4) t^4 -\zeta(6) t^6 -\zeta(8)t^8-\zeta(10)t^{10} \cdots.$$
$$\frac{\pi ti}{e^{2\pi t i}-1}-\frac{1}{2}+\frac{\pi t i}{2} +\frac{\pi t}{e^{2\pi t}-1}-\frac{1}{2}+\frac{\pi t}{2}=-2( \zeta(4)t^4+\zeta(8)t^8+\zeta(12)t^{12}+\zeta(16)t^{16}..)$$

I think you can figure out the rest…

I think that:

If we know $$\sum_{n\geq1}\frac{1}{n^4-a^4} = \frac{1}{2a^2}-\frac{\pi}{4 a^3}(\cot \pi a+\coth \pi a)$$
Then, with $a = \sqrt[4]{-1}$:
$$\sum_{n\geq1}\frac{1}{n^4+1} \approx 0.57847757966713683831802219…$$