Real zeros of $L'(s, \chi_d)$

In 1990, R. C. Baker and H. L. Montgomery conjectured that for almost all fundamental discriminants d, the derivative of the Dirichlet L-function associated to the quadratic character modulo d has around $\log\log |d|$ real zeros on the interval $[1/2, 1]$. Baker and Montgomery's motivation in studying these zeros stems from their connection to real zeros of Fekete polynomials and to sign changes of real character sums. In this talk I will present recent work that settles this conjecture (up to a small factor of $\log\log\log |d|$). This is based on a joint work with Oleksiy Klurman and Marc Munsch for the lower bound, and a more recent work joint with Kunjakanan Nath for the upper bound.