How many real zeros does a random Dirichlet series have?
Let F(σ) = P∞ n=1 Xn nσ be a random Dirichlet series where (Xn)n∈N are independent standard Gaussian random variables. We compute in a quantitative form the expected number of zeros of F(σ) in the interval [T, ∞), say EN(T, ∞), as T → 1/2 +. We also estimate higher moments and with this we derive ex...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2024 |
| País: | Brasil |
| Institución: | Universidade Federal do Rio Grande do Sul (UFRGS) |
| Repositorio: | Repositório Institucional da UFRGS |
| Idioma: | inglés |
| OAI Identifier: | oai:www.lume.ufrgs.br:10183/271776 |
| Acceso en línea: | http://hdl.handle.net/10183/271776 |
| Access Level: | acceso abierto |
| Palabra clave: | Probabilidade Series de dirichlet Random series Dirichlet series Zeros of random functions |
| Sumario: | Let F(σ) = P∞ n=1 Xn nσ be a random Dirichlet series where (Xn)n∈N are independent standard Gaussian random variables. We compute in a quantitative form the expected number of zeros of F(σ) in the interval [T, ∞), say EN(T, ∞), as T → 1/2 +. We also estimate higher moments and with this we derive exponential tails for the probability that the number of zeros in the interval [T, 1], say N(T, 1), is large. We also consider almost sure lower and upper bounds for N(T, ∞). And finally, we also prove results for another class of random Dirichlet series, e.g., when the summation is restricted to prime numbers. |
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