Period function of Maass forms from Ramanujan's lost notebook

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 $F_1\left(x\right)$. In this paper, we show that $F_1\left(x\right)$ belongs to the category of period functions as it satisfies the period relations of Maass forms in the sense of Lewis and Zagier [11]. Hence, we refer to $F_1\left(x\right)$ as the Ramanujan period function. Moreover, one of the salient aspects of the Ramanujan period function $F_1\left(x\right)$ 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 $F_1\left(x\right)$ to include a parameter s, connecting our work to the broader theory of period functions developed by Bettin and Conrey [4] and Lewis and Zagier [11]. We emphasize that Ramanujan was the first to study this function, marking the beginning of the study of period functions.