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We focus on three pages in Ramanujan's lost notebook, pages 336, 335, and332, in decreasing order of attention. On page 336, Ramanujan proposes twoidentities, but the formulas are wrong -- each is vitiated by divergent series.We concentrate on only one of the two incorrect "identities", which may havebeen devised to attack the extended divisor problem. We prove here a correctedversion of Ramanujan's claim, which contains the convergent series appearing init. The convergent series in Ramanujan's faulty claim is similar to one used byG.F. Voronoi, G.H. Hardy, and others in their study of the classical Dirichletdivisor problem. The page 335 comprises two formulas featuring doubly infiniteseries of Bessel functions, the first being conjoined with the classical circleproblem initiated by Gauss, and the second being associated with the Dirichletdivisor problem. The first and fourth authors, along with Sun Kim, have writtenseveral papers providing proofs of these two difficult formulas in differentinterpretations. In this monograph, we return to these two formulas and examinethem in more general settings.  The Voronoi summation formula appears prominently in our study. Inparticular, we generalize work of J.R. Wilton and derive an analogue involvingthe sum of divisors function $\sigma_s(n)$.  We also establish here new series and integral identities involving modifiedBessel functions and modified Lommel functions. Among other results, weestablish a modular transformation for an infinite series involving$\sigma_{s}(n)$ and modified Lommel functions. We define and discuss two newrelated classes of integral transforms, which we call Koshliakov transforms,because N.S. Koshliakov first found elegant special cases of each.

New pathways and connections in Number Theory and Analysis motivated by two incorrect claims of Ramanujan