A Ramanujan sum involving $\sinh$

This can be derived very simply using contour integration in the complex plane. Consider the contour integral

$$\pi \oint_C \frac{dz}{z^3 \sinh{(\pi z)} \sin{(\pi z)}} $$

where $C$ is a square centered at the origin of side $2 N+1$. As $N \to \infty$, one may show that the contour integral approaches zero. (Consider the magnitude of the integrand over the sides of the square.)

This means that the sum of the residues at the poles $z=\pm n$ and $z=\pm i n$ vanishes. Note that for $n \ne 0$, the residues at $z=n$ and $z=i n$ are equal to $(-1)^n/(\pi n^3 \sinh{(\pi n)})$. Also, the summand is even in $n$. Thus we have $$4 \sum_{n=1}^{\infty} \frac{(-1)^n}{n^3 \sinh{(\pi n)}} + \operatorname*{Res}_{z=0} \frac{\pi}{z^3 \sinh{(\pi z)} \sin{(\pi z)}} = 0$$

Because

$$\operatorname*{Res}_{z=0} \frac{\pi}{z^3 \sinh{(\pi z)} \sin{(\pi z)}} = \frac{\pi^3}{90} $$

the stated result follows. (NB that last residue at $z=0$ is best done using a Laurent series expansion, as the pole is of order 5.)

Suppose we seek to show that $$\sum_{n\ge 1} \frac{(-1)^{n+1}}{n^3} \frac{1}{\sinh(\pi n)} = \frac{\pi^3}{360}.$$

Using $$\frac{1}{\sinh(x)} = \frac{2}{e^x-e^{-x}} = 2\frac{e^{-x}}{1-e^{-2x}}$$ this is the same as

$$\sum_{n\ge 1} \frac{(-1)^{n+1}}{n^3} \frac{e^{-n\pi}}{1-e^{-2n\pi}} = \frac{\pi^3}{720}.$$

The sum term may be evaluated using harmonic summation techniques. Since this method has not been presented I will detail this calculation here.

Let $p$ be a positive integer and introduce

$$S(x;p) = \sum_{n\ge 1} \frac{(-1)^{n+1}}{n^{2p+1}} \frac{e^{-nx}}{1-e^{-2nx}}.$$

We will evaluate $S(\pi;p)$ using a functional equation for $S(x;p)$ that is obtained by inverting its Mellin transform.

Recall the harmonic sum identity $$\mathfrak{M}\left(\sum_{k\ge 1} \lambda_k g(\mu_k x);s\right) = \left(\sum_{k\ge 1} \frac{\lambda_k}{\mu_k^s} \right) g^*(s)$$ where $g^*(s)$ is the Mellin transform of $g(x).$

In the present case we have $$\lambda_k = \frac{(-1)^{k+1}}{k^{2p+1}}, \quad \mu_k = k \quad \text{and} \quad g(x) = \frac{e^{-x}}{1-e^{-2x}}.$$

We need the Mellin transform $g^*(s)$ of $g(x)$ which is

$$\int_0^\infty \frac{e^{-x}}{1-e^{-2x}} x^{s-1} dx = \int_0^\infty \sum_{q\ge 0} e^{- (2 q+1) x} x^{s-1} dx = \sum_{q\ge 0} \int_0^\infty e^{-(2q+1)x} x^{s-1} dx \\= \Gamma(s) \sum_{q\ge 0} \frac{1}{(2q+1)^s} = \left(1-\frac{1}{2^s}\right) \Gamma(s) \zeta(s)$$ with fundamental strip $\langle 1, \infty\rangle.$

It follows that the Mellin transform $Q(s)$ of the harmonic sum $S(x;p)$ is given by

$$Q(s) = \left(1-\frac{1}{2^s}\right) \left(1-\frac{1}{2^{s+2p}}\right) \Gamma(s) \zeta(s) \zeta(s+2p+1) \\ \text{because}\quad \sum_{k\ge 1} \frac{\lambda_k}{\mu_k^s} = \sum_{k\ge 1} \frac{(-1)^{k+1}}{k^{2p+1}} \frac{1}{k^s} = \left(1-\frac{2}{2^{s+2p+1}}\right) \zeta(s+2p+1)$$ for $\Re(s+2p+1) > 1$ or $\Re(s) \gt -2p.$

The Mellin inversion integral here is $$\frac{1}{2\pi i} \int_{3/2-i\infty}^{3/2+i\infty} Q(s)/x^s ds$$ which we evaluate by shifting it to the left for an expansion about zero.

Fortunately the trivial zeros of the two zeta function terms cancel the poles of the gamma function term. The first term cancels those at $-2m$ where $m\ge 1$ and the second one the odd ones from $-2p-3$ on, which leaves the poles at $s=1$, and $-2q-1$ with $0\le q\le p.$ It would appear there is a pole at $s=-2p$ but this is not the case since we have two simple poles among the five terms but also two zero values, making for cancelation. The pole at $s=0$ is canceled as well.

For the residue at $s=1$ we find

$$\frac{1}{2} \frac{2^{2p+1}-1}{2^{2p+1}} \times 1 \times \zeta(2p+2) \frac{1}{x} = \frac{2^{2p+1}-1}{2^{2p+2}} \frac{(-1)^{p} B_{2p+2} (2\pi)^{2p+2}}{2 (2p+2)!} \frac{1}{x} \\ = (2^{2p+1}-1) \frac{(-1)^{p} B_{2p+2} \pi^{2p+2}}{2 (2p+2)!} \frac{1}{x}.$$

The negative odd values at $s=-2q-1$ yield

$$\left(1-\frac{1}{2^{-2q-1}}\right) \left(1-\frac{1}{2^{2p-2q-1}}\right) \frac{(-1)^{2q+1}}{(2q+1)!} \zeta(-2q-1) \zeta(2p-2q) x^{2q+1} \\ = (1-2^{2q+1}) \left(1-\frac{1}{2^{2p-2q-1}}\right) \frac{1}{(2q+1)!} \frac{B_{2q+2}}{2q+2} \frac{(-1)^{p-q+1} B_{2p-2q} (2\pi)^{2p-2q}}{2(2p-2q)!} x^{2q+1} \\ = \frac{1}{2} (1-2^{2q+1}) (2^{2p-2q-1}-1) \frac{(-1)^{p-q+1}}{(2q+1)!} \frac{B_{2q+2} B_{2p-2q} \pi^{2p-2q}}{(2p-2q)! (q+1)} x^{2q+1}.$$

Shifting to $\Re(s) = -2p -3/2$ we get

$$S(x;p) = (2^{2p+1}-1) \frac{(-1)^{p} B_{2p+2} \pi^{2p+2}}{2 (2p+2)!} \frac{1}{x} \\ + \frac{1}{2} \sum_{q=0}^p (1-2^{2q+1}) (2^{2p-2q-1}-1) \frac{(-1)^{p-q+1}}{(2q+1)!} \frac{B_{2q+2} B_{2p-2q} \pi^{2p-2q}}{(2p-2q)! (q+1)} x^{2q+1} \\ + \frac{1}{2\pi i} \int_{-2p-3/2-i\infty}^{-2p-3/2+i\infty} Q(s)/x^s ds.$$

We will turn this into the promised functional equation.

Substitute $s = -2p - t$ in the remainder integral to get

$$- \frac{1}{2\pi i} \int_{3/2+i\infty}^{3/2-i\infty} \left(1-2^{2p+t}\right) \left(1-2^t\right) \Gamma(-2p-t) \zeta(-2p-t) \zeta(1-t) x^{t+2p} dt$$ which is

$$\frac{x^{2p}}{2\pi i} \int_{3/2-i\infty}^{3/2+i\infty} \left(1-2^{2p+t}\right) \left(1-2^t\right) \Gamma(-2p-t) \zeta(-2p-t) \zeta(1-t) x^{t} dt.$$

In view of the desired functional equation we now use the functional equation of the Riemann zeta function on $Q(s)$ to prove that the integrand of the last integral is in fact $(-1)^p Q(t)/\pi^{2p+2t}.$

Start with the functional equation $$\zeta(1-s) = \frac{2}{2^s\pi^s} \cos\left(\frac{\pi s}{2}\right) \Gamma(s) \zeta(s)$$ and substitute this into $Q(s)$ to obtain

$$Q(s) = \left(1-\frac{1}{2^s}\right) \left(1-\frac{1}{2^{s+2p}}\right) \frac{\zeta(1-s) 2^s \pi^s}{2\cos\left(\frac{\pi s}{2}\right)} \zeta(s+2p+1) \\ = \frac{1}{2} (2^s-1) \left(1-\frac{1}{2^{s+2p}}\right) \pi^s \frac{\zeta(s+2p+1)}{\cos\left(\frac{\pi s}{2}\right)} \zeta(1-s).$$

Apply the functional equation again (this time to $\zeta(s+2p+1)$) to get

$$Q(s) = \frac{1}{2} \frac{\pi^s}{\cos\left(\frac{\pi s}{2}\right)} (2^s-1) \left(1-\frac{1}{2^{s+2p}}\right) \frac{2}{2^{-2p-s} \pi^{-2p-s}} \cos\left(\frac{\pi (-2p-s)}{2}\right) \\ \times \Gamma(-2p-s) \zeta(-2p-s) \zeta(1-s) \\ = \frac{\pi^s}{\cos\left(\frac{\pi s}{2}\right)} (2^s-1) (2^{2p+s}-1) \pi^{2p+s} (-1)^p \cos\left(\frac{-\pi s}{2}\right) \\ \times \Gamma(-2p-s) \zeta(-2p-s) \zeta(1-s)$$

and we finally get $$Q(s) = (-1)^p \pi^{2p+2s} (1-2^s) (1-2^{2p+s}) \Gamma(-2p-s) \zeta(-2p-s) \zeta(1-s)$$

thus proving the claim.

Return to the remainder integral and re-write it as follows:

$$(-1)^p \frac{(x/\pi)^{2p}}{2\pi i} \int_{3/2-i\infty}^{3/2+i\infty} (-1)^p \pi^{2p+2t} \left(1-2^{2p+t}\right) \left(1-2^t\right) \\ \times \Gamma(-2p-t) \zeta(-2p-t) \zeta(1-t) (x/\pi^2)^{t} dt.$$

so that the fact of it being a multiple of the defining integral of $S(\pi^2/x; p)$ becomes readily apparent.

We have established the functional equation

$$\color{green} {S(x;p) = (2^{2p+1}-1) \frac{(-1)^{p} B_{2p+2} \pi^{2p+2}}{2 (2p+2)!} \frac{1}{x} \\ + \frac{1}{2} \sum_{q=0}^p (1-2^{2q+1}) (2^{2p-2q-1}-1) \frac{(-1)^{p-q+1}}{(2q+1)!} \frac{B_{2q+2} B_{2p-2q} \pi^{2p-2q}}{(2p-2q)! (q+1)} x^{2q+1} \\ + (-1)^p \left(\frac{x}{\pi}\right)^{2p} S(\pi^2/x;p).}$$

Now the value $x=\pi$ is obviously special here (fixed point) and we get for $p=2r+1$ with $r\ge 0$ ($p$ even yields a Bernoulli number identity)

$$S(\pi; 2r+1) = \sum_{n\ge 1} \frac{(-1)^{n+1}}{n^{4r+3}} \frac{e^{-n\pi}}{1-e^{-2n\pi}} = - \frac{\pi^{4r+3}}{4} (2^{4r+3}-1) \frac{B_{4r+4}}{(4r+4)!} \\ + \frac{\pi^{4r+3}}{4} \sum_{q=0}^{2r+1} (1-2^{2q+1}) (2^{4r+1-2q}-1) \frac{(-1)^{q}}{(2q+1)!} \frac{B_{2q+2} B_{4r+2-2q}}{(4r+2-2q)! (q+1)}.$$

We obtain a rational multiple of $\pi^{4r+3}.$ Scale by two to get for

$$\sum_{n\ge 1} \frac{(-1)^{n+1}}{n^3} \frac{1}{\sinh(\pi n)}, \quad \sum_{n\ge 1} \frac{(-1)^{n+1}}{n^7} \frac{1}{\sinh(\pi n)}, \\ \sum_{n\ge 1} \frac{(-1)^{n+1}}{n^{11}} \frac{1}{\sinh(\pi n)}, \quad \sum_{n\ge 1} \frac{(-1)^{n+1}}{n^{15}} \frac{1}{\sinh(\pi n)}, \quad \ldots$$

the values

$$\bbox[5px,border:2px solid #00A000]{ {\frac {{\pi }^{3}}{360}},\quad {\frac {13\,{\pi }^{7}}{453600}},\quad {\frac {4009\,{\pi }^{11}}{13621608000}},\quad {\frac {13739\,{\pi }^{15}}{4547140416000}},\quad \ldots}$$

These are dominated by the first term

$$\frac{1}{\sinh(\pi)}.$$

I realized my original proof can be shortened by exploiting symmetry.
The first three lines are unchanged: $$\frac{1}{\sinh z}=\frac{1}{z}+\sum_{m\geq 1}\left(\frac{1}{z-m\pi i}+\frac{1}{z+m\pi i}\right)(-1)^m $$ $$\frac{1}{\sinh(\pi n)}=\frac{1}{\pi n}+\frac{1}{\pi}\sum_{m\geq 1}\left(\frac{1}{n-mi}+\frac{1}{n+mi}\right)(-1)^m $$ $$\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\sinh(\pi n)}=\frac{\eta(4)}{\pi}+\frac{2}{\pi}\sum_{m\geq 1}\sum_{n\geq 1}\frac{(-1)^{n+m+1}}{n^2(n^2+m^2)}$$ then by using $2\sum_{m,n\geq 1}f(m,n) = \sum_{m,n\geq 1}f(m,n)+f(n,m)$ we immediately get $$\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\sinh(\pi n)}=\frac{\eta(4)-\eta(2)^2}{\pi}=\color{red}{\frac{\pi^3}{360}}$$ where $\eta(s)=\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^s}=(1-2^{1-s})\,\zeta(s)$ for any $s>1$.