Chowla’s non-vanishing conjecture for quadratic Dirichlet characters

Prove that for every primitive quadratic Dirichlet character \(\chi\), the central value \(L(\tfrac{1}{2},\chi)\) is non-zero.

Background

The paper begins by recalling Chowla’s conjecture concerning central values of quadratic Dirichlet LL-functions. It asserts universal, rather than statistical, non-vanishing for primitive quadratic Dirichlet characters. The surrounding discussion describes partial results giving non-vanishing for positive proportions of characters, but does not resolve the conjecture in its number-field setting.

References

Chowla has conjectured in that for all (primitive) quadratic Dirichlet characters $\chi$, the value $L\left(\frac{1}{2}, \chi \right)$ is non-zero.

— Chowla's non-vanishing conjecture over $\mathbb{F}_q(T)$  (2609.11855 - Koymans et al., 10 Sep 2026) in Section 1, Introduction