Explicit transformations of certain Lambert series

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