There are infinitely many real solutions to the equation $\tan x=x$. Denote the increasing sequence of positive solutions by $ (\lambda_n)_{n=1}^{\infty}$. I want to evaluate the sum
$$ \sum_{n=1}^{\infty} \frac{1}{\lambda_n^2} \hspace{1cm} (1) $$ (the series can be seen to converge by using a crude approximation such as $\lambda_n \approx \frac{\pi(2n+1)}{2}$).
To find the sum I'm thinking to use Hadamard's product theorem applied to a function whose roots are $\lambda_n$ and then to expand the product and compare coefficients as in Euler's evaluation of $\zeta(2)$. The Hadamard product theorem states that any entire function $f(z)$ of growth order $\alpha<2$ can be written as $$ f(z) = z^m e^{Az+B} \prod_{\rho} e^{z/\rho} \left(1-\frac{z}{\rho} \right) $$ where the product is over all the non-zero complex roots $\rho$ of $f$ repeated according to multiplicity, $m \geq 0$ is the order of the root of $f$ at $z=0$, and $A,B$ are two constants to be determined.
Now $ \tan z-z $ is not entire but it has the same non-zero roots as the function $ f(z) = \sin z - z \cos z$ which is entire. These are all real roots of order 1 and they are symmetric since the function is odd. Thus the non-zero roots of $f(z)$ are at $\pm \lambda_n$ and there is also a triple root at $z=0$. Further, the function has growth order 1, so applying Hadamard's product formula gives $$ \sin z-z \cos z = e^{Az+B} z^3 \prod_{n=1}^{\infty} e^{z/\lambda_n} \left(1-\frac{z}{\lambda_n} \right) e^{z/(-\lambda_n)} \left(1-\frac{z}{(-\lambda_n)} \right) = e^{Az+B} z^3 \prod_{n=1}^{\infty} \left(1-\frac{z^2}{\lambda_n^2} \right) $$ and we may compute $$e^B = \lim_{z \to 0} \frac{\sin z-z\cos z}{z^3} = \frac{1}{3} $$ and $A=0$ since $\frac{\sin z-z\cos z}{z^3}$ is an even function. Therefore $$ \sin z-z \cos z = \frac{z^3}{3} \prod_{n=1}^{\infty} \left(1-\frac{z^2}{\lambda_n^2} \right) \hspace{1cm} (2) $$ We may now expand the left side in a Taylor series and compare coefficients with the right side to obtain $$ \sum_{n=1}^{\infty} \frac{1}{\lambda_n^2} = \frac{1}{10} $$ as well as higher order sums such as $$ \sum_{n=1}^{\infty} \frac{1}{\lambda_n^4} = \frac{1}{350} $$ and so on. All sums of reciprocals of even powers will be rational numbers.
My questions are:
Is this solution correct?
Are there other ways of evaluating the series $(1)$ without using the Hadamard product?
Are there any general techniques (contour integration for example) available for evaluating sums of the form $$ \sum_{\rho} g(\rho) $$ where the sum is over all roots $\rho$ of an entire function $f$ and $g$ is some other analytic function for which the series converges? Thanks!
