Bailey-Zeta Limits: A $q$-Series Bridge to Dirichlet $L$-Functions and the Riemann Zeta Function
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915612635168768 |
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| author | Gurram, Mahipal |
| author_facet | Gurram, Mahipal |
| contents | We introduce a family of deformed Bailey pairs whose $q$-series, which converge in a two-step limit ($q \to 1^-$ followed by $n \to \infty$) to Dirichlet $L$-functions scaled by $1/\sqrtπ$. This construction generalizes to arbitrary bounded arithmetic progressions via character weights, providing a unified $q$-series asymptotic for $L(s,χ)$. Our approach unveils deep connections between the combinatorial machinery of Bailey chains and analytic number theory, with applications to special values like Euler-Mascheroni constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03009 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bailey-Zeta Limits: A $q$-Series Bridge to Dirichlet $L$-Functions and the Riemann Zeta Function Gurram, Mahipal General Mathematics Primary 33D15, Secondary 11M06, 11M35, 33B15 We introduce a family of deformed Bailey pairs whose $q$-series, which converge in a two-step limit ($q \to 1^-$ followed by $n \to \infty$) to Dirichlet $L$-functions scaled by $1/\sqrtπ$. This construction generalizes to arbitrary bounded arithmetic progressions via character weights, providing a unified $q$-series asymptotic for $L(s,χ)$. Our approach unveils deep connections between the combinatorial machinery of Bailey chains and analytic number theory, with applications to special values like Euler-Mascheroni constant. |
| title | Bailey-Zeta Limits: A $q$-Series Bridge to Dirichlet $L$-Functions and the Riemann Zeta Function |
| topic | General Mathematics Primary 33D15, Secondary 11M06, 11M35, 33B15 |
| url | https://arxiv.org/abs/2511.03009 |