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

TL;DR

This paper proves that for almost all fundamental discriminants, the number of real zeros of the derivative of the associated $L$-function in [1/2,1] is bounded above by a function involving iterated logarithms, nearly confirming a longstanding conjecture.

Contribution

It establishes an almost sure upper bound on the number of real zeros of $L'(s,\chi_d)$, nearly resolving the Baker-Montgomery conjecture up to a logarithmic factor.

Findings

Proves an upper bound of $( ext{log log }|d|)( ext{log log log }|d|)$ for the zeros.

Shows that almost all zeros are away from the critical point $1/2$, under certain assumptions.

Provides bounds on the size of the exceptional set of discriminants.

Abstract

In 1990, Baker and Montgomery conjectured that $L'(s,\chi_d)$ 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 $\sum_{n\leq x}\chi_d(n)$. 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 $d$, $L'(s,\chi_d)$ 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 $1/2$.