Real zeros of $L'(s, χ_d)$

TL;DR

This work addresses the location and abundance of real zeros of the derivative $L'(s,χ_d)$ for quadratic Dirichlet $L$-functions as the discriminant $d$ varies over fundamental discriminants. The authors combine (i) mean-value and orthogonality results for quadratic characters with a probabilistic random model, (ii) Selberg-type short Dirichlet polynomial approximations to $-L'/L(s,χ_d)$ valid to the right of the critical line, (iii) a discrepancy bound showing the arithmetic distribution matches the random model, and (iv) Jensen-type arguments to count zeros near the central region. Conditional on a mild hypothesis on low-lying zeros (which follows from GRH and the Katz–Sarnak one-level density), they nearly resolve Baker–Montgomery’s conjecture, proving an upper bound of order $(\,\log\log|d|)(\log\log\log|d|)$ for the number of real zeros away from $1/2$ and, in fact, that almost all zeros lie to the right of $1/2+ν(|d|)/\log|d|$. The results advance understanding of the horizontal distribution of zeros of derivatives of $L$-functions and provide evidence toward the conjectured asymptotic $\asymp \log\log|d|$ zeros in $[1/2,1]$.

Abstract

Let $ν$ be any positive function such that $ν(x)\to\infty$ as $x\to \infty$. We prove that for almost all fundamental discriminants $d$, $L'(s, χ_d)$ has at most $(\log\log |d|)( \log\log \log |d|)$ real zeros inside the interval $[1/2+ν(|d|)/\log |d|, 1]$. Combining this result with a recent work of Klurman, Lamzouri, and Munsch, shows that the number of these zeros equals $(\log\log |d|) (\log\log \log |d|)^θ$ for almost all $d$, where $|θ|\leq 1$. This comes close to proving a conjecture of Baker and Montgomery, which predicts $\asymp \log\log |d|$ real zeros of $L'(s, χ_d)$ in the interval $[1/2, 1]$, for almost all $d$. Moreover, assuming a mild hypothesis on the low lying zeros of quadratic Dirichlet $L$-functions (which follows from GRH and the one level density conjecture of Katz and Sarnak), we fully resolve the Baker-Montgomery conjecture (up to the $\log\log\log |d|$ factor). We also show, under the same hypothesis, that for almost all $d$, $100\%$ of the real zeros of $L'(s, χ_d)$ on $[1/2, 1]$ lie to the right of $1/2+ν(|d|)/\log |d|$.

Figures (2)

• Figure 1: Circles covering $[t_x, 1]$, where $t_x=1/2+\nu(x)/\log x$.
• Figure 2: Four concentric circles $\mathcal{C}_0$, $\mathcal{C}_1$, $\mathcal{C}_2$, and $\mathcal{C}_3$.

Theorems & Definitions (31)

• Conjecture 1.1: BaMo89, Baker-Montgomery
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof