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solving a heat transfer problem in cylindrical coordinates I came out with an identity, that I'm not able to prove analytically.

Let us indicate with $z_n$ the $n$-th zero of Bessel function $J_0(z)$. Is it true that

$\sum_{n=1}^{\infty}\dfrac{1}{z_n^2}=\dfrac{1}{4}$ ?

Basing on physical considerations and on numerical cross-check, I have reason to think that the equation holds, but I've not found direct evidence. Is it a known result?

Thanks in advance for replies!

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    https://arxiv.org/pdf/1305.0228v3.pdf would be of interest for you – 2017-01-17

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I am buffled since I am also getting 1/4 as a (particular) result of heat transfer problem in cylindrical coordinates, involving also roots of Bessel functions.

The 1/4 result I am getting is the Dirichlet Regularization of a sum: $$ \lim_{s->0} \sum_{n=1}^\infty (-1)^nn^{1-s}=-\frac{1}{4} $$

Probably sheer coincidence. I can't see what both sums have in common, but since both stem from Bessel roots concerning a heat transfer problem, you might try a Dirichlet regularization for your's to check.