Period Function of Maass forms from Ramanujan's Lost Notebook
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912344045518848 |
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| author | Choie, YoungJu Kumar, Rahul |
| author_facet | Choie, YoungJu Kumar, Rahul |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17284 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Period Function of Maass forms from Ramanujan's Lost Notebook Choie, YoungJu Kumar, Rahul Number Theory Classical Analysis and ODEs 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. |
| title | Period Function of Maass forms from Ramanujan's Lost Notebook |
| topic | Number Theory Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2504.17284 |