## Abstract

In the number field case, it is conjectured that the central values L(^{1}_{2} , χ) of L-functions are nonzero, where χ : (Z/mZ)^{∗} → C^{∗} is a primitive Dirichlet character with conductor m. We resolve this conjecture in the function field case by proving that there are infinitely many cyclotomic characters for which the central values of L-functions are nonzero. In detail, for a given positive integer n, we compute the mean value of L(^{1}_{2} , ηχ_{n}) and that of L(^{1}_{2} , χ_{n}) for χ_{n} ∈ O_{n}, where f is a monic irreducible polynomial in A = F_{q}[t], F_{q} is the finite field of characteristic p, χ_{n} : (A/f^{n}A)^{∗} → C^{∗} is a character with some p-power order, O_{n} is the set of all the primitive cyclotomic characters χ_{n} modulo f^{n} with p-power order, g is a monic polynomial in A that is relatively prime to f, and η : (A/gA)^{∗} → C^{∗} is a primitive even character.

Original language | English |
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Pages (from-to) | 455-468 |

Number of pages | 14 |

Journal | Proceedings of the American Mathematical Society |

Volume | 150 |

Issue number | 2 |

DOIs | |

State | Published - 2022 |

### Bibliographical note

Publisher Copyright:2021 American Mathematical Society

## Keywords

- Central value
- Cyclotomic character
- Function field
- L-function
- Mean value