A Number Field Analogue of Ramanujan's identity for $ζ(2m+1)$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908666087604224 |
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| author | Bansal, Diksha Rani Maji, Bibekananda |
| author_facet | Bansal, Diksha Rani Maji, Bibekananda |
| contents | Ramanujan's famous formula for $ζ(2m+1)$ has captivated the attention of numerous mathematicians over the years. Grosswald, in 1972, found a simple extension of Ramanujan's formula which in turn gives transformation formula for Eisenstein series over the full modular group. Recently, Banerjee, Gupta and Kumar found a number field analogue of Ramanujan's formula. In this paper, we present a new number field analogue of the Ramanujan-Grosswald formula for $ζ(2m+1)$ by obtaining a formula for Dedekind zeta function at odd arguments. We also obtain a number field analogue of an identity of Chandrasekharan and Narasimhan, which played a crucial role in proving our main identity. As an application, we generalize transformation formula for Eisenstein series $G_{2k}(z)$ and Dedekind eta function $η(z)$. A new formula for the class number of a totally real number field is also obtained, which provides a connection with the Kronceker's limit formula for the Dedekind zeta function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03660 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Number Field Analogue of Ramanujan's identity for $ζ(2m+1)$ Bansal, Diksha Rani Maji, Bibekananda Number Theory Primary 11M06, 11R42, Secondary 11R29 Ramanujan's famous formula for $ζ(2m+1)$ has captivated the attention of numerous mathematicians over the years. Grosswald, in 1972, found a simple extension of Ramanujan's formula which in turn gives transformation formula for Eisenstein series over the full modular group. Recently, Banerjee, Gupta and Kumar found a number field analogue of Ramanujan's formula. In this paper, we present a new number field analogue of the Ramanujan-Grosswald formula for $ζ(2m+1)$ by obtaining a formula for Dedekind zeta function at odd arguments. We also obtain a number field analogue of an identity of Chandrasekharan and Narasimhan, which played a crucial role in proving our main identity. As an application, we generalize transformation formula for Eisenstein series $G_{2k}(z)$ and Dedekind eta function $η(z)$. A new formula for the class number of a totally real number field is also obtained, which provides a connection with the Kronceker's limit formula for the Dedekind zeta function. |
| title | A Number Field Analogue of Ramanujan's identity for $ζ(2m+1)$ |
| topic | Number Theory Primary 11M06, 11R42, Secondary 11R29 |
| url | https://arxiv.org/abs/2407.03660 |