A Mean Value Theorem for general Dirichlet Series
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929725530701824 |
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| author | Broucke, Frederik Hilberdink, Titus |
| author_facet | Broucke, Frederik Hilberdink, Titus |
| contents | In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)= \sum_{j=1}^\infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = \sum_{n_{j}\le x}a_{j}$ satisfies $A(x)=ρx + O(x^β)$ for some $ρ>0$ and $β<1$. We prove that $\frac1T\int_0^T |f(σ+it)|^2\, dt \to \sum_{j=1}^\infty a_j^2n_j^{-2σ}$ for $σ>\frac{1+β}{2}$ and obtain an upper bound for this moment for $β<σ\le \frac{1+β}{2}$. We provide a number of examples indicating the sharpness of our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06301 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Mean Value Theorem for general Dirichlet Series Broucke, Frederik Hilberdink, Titus Number Theory Complex Variables 30B50, 11M41 In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)= \sum_{j=1}^\infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = \sum_{n_{j}\le x}a_{j}$ satisfies $A(x)=ρx + O(x^β)$ for some $ρ>0$ and $β<1$. We prove that $\frac1T\int_0^T |f(σ+it)|^2\, dt \to \sum_{j=1}^\infty a_j^2n_j^{-2σ}$ for $σ>\frac{1+β}{2}$ and obtain an upper bound for this moment for $β<σ\le \frac{1+β}{2}$. We provide a number of examples indicating the sharpness of our results. |
| title | A Mean Value Theorem for general Dirichlet Series |
| topic | Number Theory Complex Variables 30B50, 11M41 |
| url | https://arxiv.org/abs/2409.06301 |