In quantum field theory the one-loop effective action is expressed in terms of the functional determinant of the (elliptic and self-adjoint) operator of small disturbances. Since the real eigenvalues of an elliptic self-adjoint operator $L$ grow without bound, a naïve computation of its functional determinant would give, as a result, a meaningless infinite quantity.
A regularized expression for the functional determinant of the operator $L$ can be obtained by using the spectral zeta function. By denoting with $\lambda_n$, $n \in \Bbb N$, the real spectrum of $L$, which acts on suitable functions defined on a smooth compact Riemannian manifold $M$, the spectral zeta function is defined as the following sum:
$$ \zeta(s)=\sum_{n=1}^\infty \lambda_{n}^{-s} $$
where $s \in \Bbb C$.
The above series is well defined in the half-plane $\Re(s)>D/2$ where $D$ is the dimension of $M$ and can be analytically continued to the entire complex plane to a meromorphic function possessing only simple poles.
The analytically continued expression of a certain class of spectral zeta functions constructed from explicitly known spectra often contains either single or double infinite series whose terms include modified Bessel functions of the second kind $K_v(z)$.
One of the more common series that appears in applications has the form:
$$ h(s)=\sum_{m=1}^\infty m^sK_{-s}(2m). $$
What class of spectral zeta functions constructed from explicitly known spectra has analytical extension of form $h(s)$? How does the derivation of $h(s)$ go?
