Real zeros of $L^{\prime }(s,{\chi }_d)$

Abstract

In 1990, Baker and Montgomery conjectured that $L^{\prime }(s,{\chi }_d)$ has $\asymp \log \log |d|$ real zeros in the interval $[1/2,1]$ for almost all fundamental discriminants $d$. The study of these zeros was motivated by their connection to real zeros of Fekete polynomials and to sign changes of the character sums ${\sum }_{n\le x}{\chi }_d(n)$. Recent work of Klurman, Lamzouri, and Munsch shows that the number of such zeros is $\gg (\log \log |d|)/(\log \log \log \log |d|)$ for almost all $d$, thereby establishing the conjectured lower bound up to the factor $\log \log \log \log |d|$. In this paper, we prove that for almost all fundamental discriminants $d$, $L^{\prime }(s,{\chi }_d)$ has at most $(\log \log |d|)(\log \log \log |d|)$ real zeros in $[1/2,1]$, thus resolving the Baker-Montgomery conjecture up to a factor of $\log \log \log |d|$. We also give a quantitative upper bound on the exceptional set of discriminants. Furthermore, we show, conditionally on certain natural assumptions, that $100\%$ of these zeros lie away from $1/2$.

Major revision. We considerably strengthen the main results and in particular, we now resolve the Baker-Montgomery conjecture unconditionally (up to a $\log \log \log x$ factor). The proof introduces new ideas, and the structure and exposition have been reorganised accordingly. 30 pages, 2 figures