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

TL;DR

This paper revisits and corrects Ramanujan's divergent series claims, generalizes classical formulas involving Bessel functions and divisor sums, and introduces new integral transforms called Koshliakov transforms.

Contribution

It corrects Ramanujan's divergent series claims, generalizes Voronoi and related formulas, and introduces novel Koshliakov transforms in number theory and analysis.

Findings

Corrected Ramanujan's divergent series identities.

Derived new series and integral identities involving Bessel and Lommel functions.

Established a modular transformation for divisor sum series.

Abstract

We focus on three pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong -- each is vitiated by divergent series. We concentrate on only one of the two incorrect "identities", which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G.F. Voronoi, G.H. Hardy, and others in their study of the classical Dirichlet divisor problem. The page 335 comprises two formulas featuring doubly infinite series of Bessel functions, the first being conjoined with the classical circle problem initiated by Gauss, and the second being associated with the Dirichlet divisor problem. The first and fourth authors, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this monograph, we return to these two formulas and examine them in more general settings. The Voronoi summation formula appears prominently in our study. In particular, we generalize work of J.R. Wilton and derive an analogue involving the sum of divisors function $\sigma_s(n)$. We also establish here new series and integral identities involving modified Bessel functions and modified Lommel functions. Among other results, we establish a modular transformation for an infinite series involving $\sigma_{s}(n)$ and modified Lommel functions. We define and discuss two new related classes of integral transforms, which we call Koshliakov transforms, because N.S. Koshliakov first found elegant special cases of each.