On a Question of Alan Baker Over Arbitrary Number Fields

Abstract

For any positive integer q, it is a question of Baker whether the numbers L(1,χ), where χ runs over the non‐trivial characters mod q, are linearly independent over double-struckQ. In this work, we consider the linear independence of the L(1,χ) values over number fields where χ runs over the non‐trivial characters modulo a prime p. The dimension over double-struckQ and over some specific number fields is known, thanks to a result of Baker, Birch and Wirsing. But little is known about the dimension over a general number field. In this note, we study the distribution and growth of these dimensions over families of number fields not covered by the Baker–Birch–Wirsing theorem. One of the crucial ingredients is a celebrated result of Linnik about the least prime in an arithmetic progression. Some of these dimension calculations are linked to Fermat and Sophie Germain primes.