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Explicit transformations of certain Lambert series

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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.

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Notes

  1. Koshliakov inadvertently missed the factor \(\pi \) in front on the right-hand side.

  2. It will be shown later that when \(\mu \ne -\nu \), this result actually holds for \(\nu \in {\mathbb {C}}\backslash \left( {\mathbb {Z}}\backslash \{0\}\right) \), \(Re (\mu )>-1/2\), \(Re (\nu )>-1/2\) and \(Re (\mu +\nu )>-1/2\); otherwise, it holds for \(-1/2<Re (\nu )<1/2\).

  3. One might as well take x in the definition of \({\mathscr {G}}_{\nu }(x, w)\) to be such that \(-\pi<\arg (x)<\pi \), thereby having analyticity in x as well. However, in this paper, we will be working with \(x>0\) only.

  4. For the definition of a resultant of two kernels, see [50].

  5. Ramanujan’s formula is actually valid for any complex \(\alpha , \beta \) such that \(Re (\alpha )>0, Re (\beta )>0\) and \(\alpha \beta =\pi ^2\).

  6. The condition for the validity of the Mellin transform given in this paper, namely \(-Re (w)\pm Re (\nu )<Re (s)<\frac{1}{4}\), is too restrictive. Here, we extend the region of validity.

  7. This integral evaluation was obtained by Koshliakov [60, Equation (13)].

  8. This lemma is valid even if k is complex such that \(Re (k)>0\).

  9. This result is actually true for all \(z\in {\mathbb {C}}\). Note that at any non-positive real number z, the right-hand side has a removable singularity.

  10. In the statement of their theorem, the \(1/\pi ^2\) appearing in front of the summation on the right-hand side should be \(1/\pi \).

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Acknowledgements

The authors sincerely thank the referees for giving very nice suggestions which improved the exposition of the paper. They would also like to thank Olivia da Costa Maya, Professors Alexandru Zaharescu, Lin Jiu and Gaurav Dwivedi, respectively, for sending them the copies of references [17, 25, 53, 64]. They also thank Becky Burner, a library staff at the University of Illinois at Urbana-Champaign, for procuring the copies of [20, 82], Dr. T. S. Kumbar, the librarian at IIT Gandhinagar for obtaining a copy of [18] and Suresh Kumar, the librarian at the Harish-Chandra Research Institute, for arranging a copy of [19]. The first author’s research was supported by the CRG grant CRG/2020/002367. He sincerely thanks SERB for the support. The third author’s research was supported by the Grant IBS-R003-D1 of the IBS-CGP, POSTECH, South Korea and by IIT Gandhinagar. He sincerely thanks both the institutes for the support.

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  1. Rahul Kumar

    Present address: Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, 37673, Republic of Korea

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  1. Discipline of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, Gujarat, 382355, India

    Atul Dixit & Rahul Kumar

  2. Computing and Information Services, Harvey Mudd College, 301 Platt Boulevard, Claremont, CA, 91711, USA

    Aashita Kesarwani

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Dixit, A., Kesarwani, A. & Kumar, R. Explicit transformations of certain Lambert series. Res Math Sci 9, 34 (2022). https://doi.org/10.1007/s40687-022-00331-5

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