An asymptotic expansion for a Lambert series associated to Siegel cusp forms of degree $n$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916262199689216 |
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| author | Babita Jha, Abhash Kumar Maji, Bibekananda Pal, Manidipa |
| author_facet | Babita Jha, Abhash Kumar Maji, Bibekananda Pal, Manidipa |
| contents | Utilizing inverse Mellin transform of the symmetric square $L$-function attached to Ramanujan tau function, Hafner and Stopple proved a conjecture of Zagier, which states that the constant term of the automorphic function $y^{12}|Δ(z)|^2$ i.e., the Lambert series $y^{12}\sum_{n=1}^\infty τ(n)^2 e^{-4 πn y}$ can be expressed in terms of the non-trivial zeros of the Riemann zeta function. This study examines certain Lambert series associated to Siegel cusp forms of degree $n$ twisted by a character $χ$ and observes a similar phenomenon. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_17205 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An asymptotic expansion for a Lambert series associated to Siegel cusp forms of degree $n$ Babita Jha, Abhash Kumar Maji, Bibekananda Pal, Manidipa Number Theory Primary 11M06, 11M26, 11F46, Secondary 11N37 Utilizing inverse Mellin transform of the symmetric square $L$-function attached to Ramanujan tau function, Hafner and Stopple proved a conjecture of Zagier, which states that the constant term of the automorphic function $y^{12}|Δ(z)|^2$ i.e., the Lambert series $y^{12}\sum_{n=1}^\infty τ(n)^2 e^{-4 πn y}$ can be expressed in terms of the non-trivial zeros of the Riemann zeta function. This study examines certain Lambert series associated to Siegel cusp forms of degree $n$ twisted by a character $χ$ and observes a similar phenomenon. |
| title | An asymptotic expansion for a Lambert series associated to Siegel cusp forms of degree $n$ |
| topic | Number Theory Primary 11M06, 11M26, 11F46, Secondary 11N37 |
| url | https://arxiv.org/abs/2405.17205 |