I am interested in :
$$ \sum_{n=1}^{\infty}(-1)^n\frac{x^{2n}\ \zeta{(2n)}}{(2n)!} = ??? $$
This could possibly be a Bessel function of some kind, because of the way the Riemann zeta function of even integer arguments can be defined using Bernoulli numbers and even argument factorials.
It can be easily assessed that, the given sum is the expanded and simplified form of the sum, $ S=\lim_{k \to \infty} \sum_{r=1}^{k} (\cos \frac{x}{r}-1) $. $S=-2\lim_{k \to \infty} \sum_{r=1}^{k} \sin^2 \frac{x}{2r} $. This is a convergent series for any $ \mid x \mid < \infty $.
It can be evaluated using Euler Maclaurin series or other summation evaluation methods but a closed form solution of this sum in elementary finite number of functions, i feel, cannot be given.