Explicit transformations of certain Lambert series

Abstract

An exact transformation, which we call the 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 nonzero 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 \(SL _{2}\left( {\mathbb {Z}}\right) \), 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 Voronoï 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.