Evaluate $\sum _{n=1}^{\infty } \sin \left(\pi \sqrt{n^2+1}\right)$

Question

How can one prove $$\sum _{n=1}^{\infty } \sin \left(\pi \sqrt{n^2+1}\right)=-\frac{1}{2}\pi Y_1(\pi )-\int_0^{\infty } \exp ^{\frac{\pi}{2} \left(t-t^{-1}\right)} (\theta(2 \pi t)-1) \, dt$$ Here $\theta$ denotes Theta function of the third kind. Correpsonding cosine case is solved here but not so helpful. Any help will be appreciated.

Answer 1

Let's see: $$ \sin\left(\pi\sqrt{n^2+1}\right)=\sin\left(\pi n+\frac{\pi}{n+\sqrt{n^2+1}}\right)=(-1)^n\sin\left(\frac{\pi}{n+\sqrt{n^2+1}}\right) $$ hence the series is convergent by Leibniz' test. The collateral series $$ S(m)=\sum_{n\geq 1}(-1)^n\left(\frac{1}{n+\sqrt{n^2+1}}\right)^{2m+1} $$ are interesting objects, related to some series due to Ramanujan. By the (inverse) Laplace transform $$ S(0)=\sum_{n\geq 1}\frac{(-1)^n}{n+\sqrt{n^2+1}}=\int_{0}^{+\infty}\frac{J_1(s)}{s}\sum_{n\geq 1}(-1)^n e^{-ns}\,ds=-\int_{0}^{+\infty}\frac{J_1(s)}{s(e^s+1)}\,ds $$ and by the integral representation for the Bessel function $J_1$ $$ \frac{J_1(s)}{s}=\frac{1}{\pi}\int_{0}^{\pi}\cos(s\cos\theta)\sin^2(\theta)\,d\theta $$ such that $$ S(0) = -\frac{1}{2\pi}\cdot\Re\int_{0}^{\pi}\psi\left(\frac{2+i\cos\theta}{2}\right)-\psi\left(\frac{1+i\cos\theta}{2}\right)\,d\theta.$$ These real parts of digamma functions are extremely well-behaved on $[0,\pi]$, hence any numerical integration algorithm is able to find $S(0)\approx -0.271597$ with arbitrary accuracy. The same approach can be applied to $S(1),S(2),\ldots$ and the sequence $\{S(n)\}_{n\geq 0}$ roughly converges to zero like $\frac{1}{(1+\sqrt{2})^{2n}}$, so it is sufficient to compute $S(n)$ up to a small $n$ with good accuracy, then invoke interpolation to approximate $$\sum_{n\geq 1}\sin(\pi\sqrt{n^2+1})=\sum_{m\geq 0}\frac{\pi^{2m+1}(-1)^m}{(2m+1)!}S(m)\approx \color{red}{-0.566582}.$$

Answer 2

We can use the Poisson summation formula after a transformation to obtain the quoted result. Defining \begin{equation} I(a)=\sum _{n=1}^{\infty } \sin \left(\pi \sqrt{n^2+a^2}\right) \end{equation} we have \begin{equation} I'(a)=\pi a\sum _{n=1}^{\infty } \frac{\cos \left(\pi \sqrt{n^2+a^2}\right)}{\sqrt{n^2+a^2}} \end{equation} For even functions, the Poisson summation formula reads \begin{equation} \sum _{n=1}^{\infty }f(n)=\sum _{k=1}^{\infty }\hat f(k)+\frac{1}{2}\left( \hat f(0)-f(0) \right) \end{equation} where \begin{equation} \hat f(k)=2\int_0^\infty f(t)\cos(2\pi kt)\,dt \end{equation} Using the tabulated integral (3.876.2) in G&R, with $a>0$, \begin{equation} \int_0^\infty \frac{\cos \left(\pi \sqrt{t^2+a^2}\right)}{\sqrt{t^2+a^2}}\cos(2\pi kt)\,dt= \begin{cases} K_0\left( \pi a\sqrt{4k^2-1} \right) & \text{for } k>1/4\\ -\frac{\pi}{2}Y_0\left(\pi a \right) & \text{for } k<1/4 \end{cases} \end{equation} Then \begin{equation} I'(a)=2\pi a\sum _{k=1}^{\infty } K_0\left( \pi a\sqrt{4k^2-1} \right) -\frac{\pi^2 a}{2}Y_0\left(\pi a \right)-\frac{\pi}{2}\cos \pi a \end{equation} From here, noticing that $I(0)=0$, we can integrate by exchanging summation and integration \begin{align} I(s)&=2\pi \sum _{k=1}^{\infty }\int_0^s K_0\left( \pi a\sqrt{4k^2-1} \right)a\,da -\frac{\pi^2}{2}\int_0^s Y_0\left(\pi a \right)a\,da-\frac{\pi}{2}\int_0^s \cos \pi a\,da\\ &=-2s\sum _{k=1}^{\infty }\frac{K_1\left( \pi s\sqrt{4k^2-1} \right)}{\sqrt{4k^2-1}}-\frac{\pi}{2}sY_1\left( \pi s \right)-\frac{1}{2}\sin \pi s \end{align} We chose an integral representation for the Bessel function DLMF \begin{equation} K_{1}\left(z\right)=\frac{z}{4}\int_{0}^{\infty}\exp \left(-t-\frac{z^{2}}{4t}\right)\frac{\mathrm{d}t}{t^2} \end{equation} The above series reads \begin{align} -2s\sum _{k=1}^{\infty }\frac{K_1\left( \pi s\sqrt{4k^2-1} \right)}{\sqrt{4k^2-1}}&=-\frac{\pi s}{2}\sum _{k=1}^{\infty }\int_0^\infty e^{-t+\frac{\pi^2s^2}{4t}-\frac{\pi^2s^2k^2}{t}}\frac{\mathrm{d}t}{t^2}\\ &=-\frac{\pi s^2}{2}\int_0^\infty e^{-t+\frac{\pi^2s^2}{4t}}\left( \sum _{k=1}^{\infty }e^{-\frac{\pi^2s^2k^2}{t}} \right)\frac{\mathrm{d}t}{t^2}\\ &=-\frac{\pi s^2}{4}\int_0^\infty e^{-t+\frac{\pi^2s^2}{4t}}\left( f\left(\frac{\pi^2s^2}{t} \right)-1 \right)\frac{\mathrm{d}t}{t^2}\\ &=-\frac{s^3}{2}\int_0^\infty e^{\frac{\pi s}{2}\left( u-\frac{ 1}{u} \right)}\left(f\left( 2\pi su \right)-1 \right)\mathrm{d}u \end{align} Where $f(\pi^2 a^2/t)=\theta_3(0,e^{-\pi^2 a^2/t})$ where $\theta_3$ is the Theta function of the third kind DLMF. The latter expression was obtained by changing $t=\pi s/(2u)$.

Finally, \begin{equation} \sum _{n=1}^{\infty } \sin \left(\pi \sqrt{n^2+s^2}\right)=-\frac{s^3}{2}\int_0^\infty e^{\frac{\pi s}{2}\left( u-\frac{ 1}{u} \right)}\left(f\left( 2\pi su \right)-1 \right)\mathrm{d}u-\frac{\pi}{2}sY_1\left( \pi s \right)-\frac{1}{2}\sin \pi s \end{equation} which seems to be numerically correct.