Non-vanishing of Dirichlet L-functions in Galois orbits

TL;DR

This paper extends non-vanishing results of Dirichlet L-functions at the central point to characters within Galois orbits, especially for prime power moduli, revealing new non-vanishing proportions in these algebraic structures.

Contribution

It proves non-vanishing proportions for Dirichlet L-functions within Galois orbits of characters, including those of fixed order and intermediate Galois groups, for prime power moduli.

Findings

At least one third of primitive characters have non-vanishing L-functions.

Non-vanishing proportion is established for characters of any fixed order.

Positive non-vanishing proportion in orbits generated by larger Galois groups.

Abstract

A well known result of Iwaniec and Sarnak states that for at least one third of the primitive Dirichlet characters to a large modulus q, the associated L-functions do not vanish at the central point. When q is a large power of a fixed prime, we prove the same proportion already among the primitive characters of any given order. The set of primitive characters modulo q of a given order can be described as an orbit under the action of the Galois group of the corresponding cyclotomic field. We also prove a positive proportion of nonvanishing within substantially shorter orbits generated by intermediate Galois groups as soon as they are larger than roughly the square-root of the prime-power conductor.