  An exact transformation, which we call the \emph{master identity}, is
obtained for the first time for the series
$\sum_{n=1}^{\infty}\sigma_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$.
New modular-type transformations when $a$ is a non-zero even integer are
obtained as its special cases. The precise obstruction to modularity is
explicitly seen in these transformations. These include a novel companion to
Ramanujan's famous formula for $\zeta(2m+1)$. The Wigert-Bellman identity
arising from the $a=0$ case of the master identity is derived too. When $a$ is
an odd integer, the well-known modular transformations of the Eisenstein series
on $\textup{SL}_{2}\left(\mathbb{Z}\right)$, that of the Dedekind eta function
as well as Ramanujan's formula for $\zeta(2m+1)$ are derived from the master
identity. The latter identity itself is derived using Guinand's version of the
Vorono\"{\dotlessi} summation formula and an integral evaluation of
N.~S.~Koshliakov involving a generalization of the modified Bessel function
$K_{\nu}(z)$. Koshliakov's integral evaluation is proved for the first time. It
is then generalized using a well-known kernel of Watson to obtain an
interesting two-variable generalization of the modified Bessel function. This
generalization allows us to obtain a new modular-type transformation involving
the sums-of-squares function $r_k(n)$. Some results on functions
self-reciprocal in the Watson kernel are also obtained.
