Riemann-Type Functional Equations -- Julia Line and Counting Formulae --
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866914876601925632 |
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| author | Sourmelidis, Athanasios Steuding, Jörn Suriajaya, Ade Irma |
| author_facet | Sourmelidis, Athanasios Steuding, Jörn Suriajaya, Ade Irma |
| contents | We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number $a\neq0$ and a function from the Selberg class $\mathcal{L}$, we prove a Riemann-von Mangoldt formula for the number of a-points of the $Δ$-factor of the functional equation of $\mathcal{L}$ and an analog of Landau's formula over these points. From the last formula we derive that the ordinates of these $a$-points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of $\mathcal{L}(s)$ taken at these points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_10692 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Riemann-Type Functional Equations -- Julia Line and Counting Formulae -- Sourmelidis, Athanasios Steuding, Jörn Suriajaya, Ade Irma Number Theory Complex Variables 11M06, 30D35 We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number $a\neq0$ and a function from the Selberg class $\mathcal{L}$, we prove a Riemann-von Mangoldt formula for the number of a-points of the $Δ$-factor of the functional equation of $\mathcal{L}$ and an analog of Landau's formula over these points. From the last formula we derive that the ordinates of these $a$-points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of $\mathcal{L}(s)$ taken at these points. |
| title | Riemann-Type Functional Equations -- Julia Line and Counting Formulae -- |
| topic | Number Theory Complex Variables 11M06, 30D35 |
| url | https://arxiv.org/abs/2011.10692 |