Truncated product representations for L-functions in the hyperelliptic ensemble
| Autor: | Steven M. Gonek, Julio Andrade, Jon P Keating |
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| Rok vydání: | 2019 |
| Předmět: |
11G20
14G10 11M50 General Mathematics zeros of L-function Modular form random matrix theory 01 natural sciences 0103 physical sciences FOS: Mathematics hybrid formula Number Theory (math.NT) 0101 mathematics 010306 general physics Mathematics Mathematics - Number Theory Mathematical society 010102 general mathematics hyperelliptic curve 16. Peace & justice function fields Algebra Finite field Product (mathematics) Scheme (mathematics) finite fields Hyperelliptic curve Random matrix |
| Zdroj: | Andrade, J C, Gonek, S M & Keating, J 2018, ' Truncated Product Representations for L-Functions in the Hyperelliptic Ensemble ', Mathematika, vol. 64, no. 1, pp. 137-158 . https://doi.org/10.1112/S0025579317000407 |
| DOI: | 10.1112/s0025579317000407 |
| Popis: | We investigate the approximation of quadratic Dirichlet $L$-functions over function fields by truncations of their Euler products. We first establish representations for such $L$-functions as products over prime polynomials times products over their zeros. This is the hybrid formula in function fields. We then prove that partial Euler products are good approximations of an $L$-function away from its zeros, and that, when the length of the product tends to infinity, we recover the original $L$-function. We also obtain explicit expressions for the arguments of quadratic Dirichlet $L$-functions over function fields and for the arguments of their partial Euler products. In the second part of the paper we construct, for each quadratic Dirichlet $L$-function over a function field, an auxiliary function based on the approximate functional equation that equals the $L$-function on the critical line. We also construct a parametrized family of approximations of these auxiliary functions, prove the Riemann hypothesis holds for them, and that their zeros are related to those of the associated $L$-function. Finally, we estimate the counting function for the zeros of this family of approximations, show that these zeros cluster near those of the associated $L$-function, and that, when the parameter is not too large, almost all the zeros of the approximations are simple. 23 pages |
| Databáze: | OpenAIRE |
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