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\title[Real zeros of L-function derivatives]{Real zeros of L-function derivatives:\\density-one asymptotics in arithmetic families}
\author{Han Yuhyun}
\date{October 14, 2026}
\subjclass[2020]{Primary 11M06, 11M41; Secondary 60G15, 60G55}
\keywords{real zeros, derivatives of L-functions, random Euler products, quadratic twists, cubic fields, Gaussian processes}
\begin{document}
\begin{abstract}
We study real zeros of the derivative of an uncompleted $L$-function on $[1/2,1]$.
The limiting stationary process has covariance $\sech^2((t-s)/2)$ after the change of variables $\sigma=1/2+e^{-t}$.
Its nonzero-level Rice intensity gives the constant $e^{-1/2}/(\pi\sqrt2)$ for quadratic Dirichlet characters, a specified family of quadratic Hecke characters over $\Q(i)$, and the unweighted level-one holomorphic family; the resulting density-one asymptotics are unconditional.
For a fixed holomorphic modular form and its quadratic twists, and for non-Galois cubic Artin $L$-functions, we obtain the same constant under the respective family GRH.
The real cyclic cubic augmentation instead has constant $1/(\pi\sqrt2)$.
In these three GRH families a uniform quantile estimate controls all three counts: distinct real zeros, zeros with multiplicity, and strict sign changes.
For an individual complex cubic character, the derivative has $o(\log\log q)$ real zeros in density under cubic-character GRH.
We also prove a uniform mean asymptotic and exponential concentration for the random Euler model.
\end{abstract}
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\input{sections/introduction}
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\input{sections/numerics}
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