Period Function of Maass forms from Ramanujan's Lost Notebook
YoungJu Choie, Rahul Kumar
TL;DR
The paper identifies Ramanujan's function $\\mathcal{F}_1(x)$, derived from $\\mathscr{F}_1(x)$, as a Maass period function with spectral parameter $s=\tfrac{1}{2}$, thereby placing it in the Lewis–Zagier framework. It proves that $\\mathcal{F}_1$ is a Hecke eigenform on the space of periods with eigenvalues $\\sqrt{n}\,d(n)$, and reveals its appearance in Kronecker limit formulas for partial zeta functions of real quadratic fields. A generalized family $\\mathcal{F}_s$ is developed, connecting to Bettin–Conrey and expanding the period-function picture beyond the original Ramanujan case. The work also links $\\mathcal{F}_1$ to Eisenstein series of weight $1$, provides special rational values, and situates Ramanujan's insights at the origin of modern period-function theory, with several explicit arithmetic applications. Overall, the results illuminate deep connections among Ramanujan’s sums, Maass forms, Hecke actions, and zeta-function limit phenomena, forming a bridge between classical and modern automorphic analysis.
Abstract
The Lost Notebook of Ramanujan contains a number of beautiful formulas, one of which can be found on its page 220. It involves an interesting function, which we denote as $\mathcal{F}_1(x)$. In this paper, we show that $\mathcal{F}_1(x)$ belongs to the category of period functions as it satisfies the period relations of Maass forms in the sense of Lewis and Zagier \cite{lz}. Hence, we refer to $\mathcal{F}_1(x)$ as the \emph{Ramanujan period function}. Moreover, one of the salient aspects of the Ramanujan period function $\mathcal{F}_1(x)$ that we found out is that it is a Hecke eigenfunction under the action of Hecke operators on the space of periods. We also establish that it naturally appears in a Kronecker limit formula of a certain zeta function, revealing its connections to various topics. Finally, we generalize $\mathcal{F}_1(x)$ to include a parameter $s,$ connecting our work to the broader theory of period functions developed by Bettin and Conrey \cite{bc} and Lewis and Zagier \cite{lz}. We emphasize that Ramanujan was the first to study this function, marking the beginning of the study of period functions.