Chowla’s conjecture on central values of primitive Dirichlet L-functions

Determine whether every primitive Dirichlet character \(\chi\) satisfies \(L(1/2,\chi)\neq 0\), as asserted by Chowla’s conjecture.

Background

The paper places its ray class nonvanishing theorem in the context of the classical nonvanishing problem for Dirichlet $L$-functions. Chowla conjectured that the central value $L(1/2,\chi)$ is nonzero for every primitive Dirichlet character. This universal nonvanishing assertion is substantially stronger than the positive-proportion result established in the paper and is presented as a conjectural statement rather than a theorem.

The authors then recall that Balasubramanian and Murty proved the first positive proportion of nonvanishing for Dirichlet characters, while Iwaniec and Sarnak obtained a proportion of $1/3$. These results provide context for the paper’s analogous positive-proportion theorem for primitive ray class characters over imaginary quadratic fields.