Special values of derivatives of certain $L$-functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913479137427456 |
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| author | Chatterjee, Tapas Dhillon, Sonika |
| author_facet | Chatterjee, Tapas Dhillon, Sonika |
| contents | In this paper we address the question of non-vanishing of $L'(0,f)$ where $f$ is an algebraic valued periodic function. In 2011, Gun, Murty and Rath studied the nature of special values of the derivatives of even Dirichlet-type functions and proved that it can be either zero or transcendental. Here for some special cases we characterize the set of functions for which $L'(0,f)$ is zero or transcendental. Using a theorem of Ramachandra about multiplicative independence of cyclotomic units we also provide some non-trivial examples of functions where $L'(0,f)$ is zero. Finally, assuming Schanuel's conjecture we derive the algebraic independence of special values of derivatives of $L$-functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_13737 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Special values of derivatives of certain $L$-functions Chatterjee, Tapas Dhillon, Sonika Number Theory 11J81, 11J86, 11M06, 11J91 In this paper we address the question of non-vanishing of $L'(0,f)$ where $f$ is an algebraic valued periodic function. In 2011, Gun, Murty and Rath studied the nature of special values of the derivatives of even Dirichlet-type functions and proved that it can be either zero or transcendental. Here for some special cases we characterize the set of functions for which $L'(0,f)$ is zero or transcendental. Using a theorem of Ramachandra about multiplicative independence of cyclotomic units we also provide some non-trivial examples of functions where $L'(0,f)$ is zero. Finally, assuming Schanuel's conjecture we derive the algebraic independence of special values of derivatives of $L$-functions. |
| title | Special values of derivatives of certain $L$-functions |
| topic | Number Theory 11J81, 11J86, 11M06, 11J91 |
| url | https://arxiv.org/abs/2408.13737 |