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An open question on convergence of a series

This is an open question for all of you that you are taking glances at my blog. I do not have a solution and I am still searching for one.

Examine if the series

    \[\mathcal{S} = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{\sin (\sin (mn))}{n^2+m^2}\]


Some background: The series appeared in 2014 in several fora including math.stackexchange.com as well as mathematica.gr .  As you can see in the link there is no answer but the comments made by the users suggest that the series converges to \frac{1}{2}.

Wolfram Alpha , on the other hand , says the series diverges. But Wolfram Alpha is just a computer program and it is not to be taken for granted since computers can be erroneous from time to time.

So, you’re more than welcome to share your thoughts with us if you come up with something worth sharing.



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