Chowla's conjecture postulates that the L-function of a quadratic character over Q should never vanish at s=1/2. At present we know that this holds for a positive proportion of characters, thanks to Soundararajan. We also know that over function fields Fq(T) the naive analogue of Chowla's conjecture is false: Li has constructed infinitely many quadratic characters with L-function vanishing at 1/2. The refined form of Chowla's conjecture postulates that the non-vanishing should hold with probability 1. This statement is grounded in the Katz--Sarnak random matrix heuristics and has seen partial evidence thanks to the works of Florea,Florea--David--Lalin, Ellenberg--Li--Shusterman, where it was established for each q a positive proportion of non-vanishing (with proportions getting better as q goes to infinity). In this talk I will discuss some of the ingredients of an upcoming joint work with Koymans and Shusterman, where we establish that the refined form of Chowla's conjecture holds for each fixed q congruent to 3 modulo 4.

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