Moments of Higher derivatives of the logarithmic derivative of Dirichlet $L$-functions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915484125888512 |
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| author | Ghosh, Samprit |
| author_facet | Ghosh, Samprit |
| contents | Let $χ$ be a non-principal Dirichlet character and $L(s, χ)$ be the associated Dirichlet $L$-function. Let us use $\mathcal{L}(s,χ)$ to denote its logarithmic derivative $L'(s, χ)/L(s, χ)$. We first prove some arithmetic formulas for higher derivatives $\mathcal{L}^{(r)}(1,χ)$. We then investigate their moments. We study the average of $P^{(a,b)}(\mathcal{L}^{(r)}(1,χ))$ as $χ$ runs over all non-principal Dirichlet characters with a given large prime conductor $m$, where $P^{(a,b)}(z) = z^a \overline{z}^b$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06390 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moments of Higher derivatives of the logarithmic derivative of Dirichlet $L$-functions Ghosh, Samprit Number Theory Let $χ$ be a non-principal Dirichlet character and $L(s, χ)$ be the associated Dirichlet $L$-function. Let us use $\mathcal{L}(s,χ)$ to denote its logarithmic derivative $L'(s, χ)/L(s, χ)$. We first prove some arithmetic formulas for higher derivatives $\mathcal{L}^{(r)}(1,χ)$. We then investigate their moments. We study the average of $P^{(a,b)}(\mathcal{L}^{(r)}(1,χ))$ as $χ$ runs over all non-principal Dirichlet characters with a given large prime conductor $m$, where $P^{(a,b)}(z) = z^a \overline{z}^b$. |
| title | Moments of Higher derivatives of the logarithmic derivative of Dirichlet $L$-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.06390 |