\section{Introduction and statements}
For a real entire $L$-function $L$, write $R_L^{\dist}$ and $R_L^{\mult}$ for the number of zeros of $L'$ on the \emph{closed} real interval $[1/2,1]$, without and with complex-analytic multiplicity. Write $S_L$ for the number of strict sign changes of $L'$ on $(1/2,1)$. All derivatives in this paper are derivatives of the uncompleted function. These conventions matter at the central point and for families with root number $-1$.

The phrase ``in density'' always refers to the original family measure. Thus $R_L/H_L\to c$ in density means that, for every fixed $\eta>0$, the proportion with $|R_L/H_L-c|>\eta$ tends to zero. This does not assert an asymptotic for every member, or an asymptotic for the family mean of the unbounded count.

Our constants are
\begin{equation}\label{intro:constants}
 \cBM=\frac{e^{-1/2}}{\pi\sqrt2},\qquad
 c(m,v)=\frac{e^{-m^2/(2v)}}{\pi\sqrt2}\quad(v>0).
\end{equation}
The parameters $m$ and $v$ are, respectively, the weighted mean of the prime-square logarithmic coefficient and the weighted variance of the prime coefficient. They are defined precisely in the Euler-model section. The variance of the derivative of a stationary Gaussian process with covariance $\sech^2(t/2)$ is $1/2$. Its crossings of the level $-m/\sqrt v$ have intensity $c(m,v)$. This is why changing the drift from $1/2$ to $-1/2$ preserves the constant, whereas removing the drift changes it.

\subsection{Unconditional arithmetic asymptotics}
Let $\mathcal F(X)$ be the set of fundamental discriminants $D$ with $1<|D|\le X$, of both signs, and let $\chi_D$ be the primitive quadratic character of conductor $|D|$.
\begin{theorem}[Quadratic Dirichlet family, unconditional]\label{intro:quadratic}
For each fixed $\eta>0$, all but $o_\eta(X)$ elements of $\mathcal F(X)$ satisfy, simultaneously,
\[
 \max_{C\in\{R^{\dist},R^{\mult},S\}}
 \left|\frac{C_{L(s,\chi_D)}}{\log\log|D|}-\cBM\right|\le\eta.
\]
No hypothesis on zeros or central values is imposed.
\end{theorem}

The full signed family in this theorem requires an arithmetic argument at the prime $2$. A theorem for the subfamily $D=8d$ cannot simply be relabelled as a theorem for all fundamental discriminants. We give the complete extension of the mollified estimate, including the other residue classes and the cancellation of the paired primary terms, in the unconditional-family section.

For the Gaussian-integer family put $\pi=1+i$. Let $d\in\Z[i]$ range over elements that are prime to $2$ and squarefree as ideals, with $X/2<N(d)\le X$. Use the primitive quadratic Hecke character $\chi_{\pi^5d}$. The finite conductor has norm $32N(d)$, and the normalization used for the completion has conductor parameter $Q_d=128N(d)$. Each character occurs with exactly the two labels $d,-d$.
\begin{theorem}[Quadratic Hecke family over $\Q(i)$, unconditional]\label{intro:gaussian}
For the preceding family, the three counts $R^{\dist},R^{\mult},S$ of $L'(s,\chi_{\pi^5d})$ each equal
\[
 (\cBM+o(1))\log\log Q_d
\]
in density. More explicitly, at each fixed relative tolerance $\eta>0$, the exceptional number of labels is $o_\eta(X)$. The same statement holds when each character is counted once and for cumulative norm cutoffs.
\end{theorem}
This is a specified quadratic Hecke family, rather than all Hecke characters over $\Q(i)$.

Let $\mathcal H_k$ be the set of normalized level-one holomorphic Hecke eigenforms of even weight $k$, each counted once. The $L$-function is normalized to have central point $1/2$.
\begin{theorem}[All even weights at level one, unconditional]\label{intro:weight}
As even $k\to\infty$, for each fixed $\eta>0$, all but $o_\eta(|\mathcal H_k|)$ forms satisfy
\[
 \max_{C\in\{R^{\dist},R^{\mult},S\}}
 \left|\frac{C_{L(s,f)}}{\log\log(k^2)}-\cBM\right|\le\eta.
\]
The measure is unweighted. Both residue classes of even $k$ modulo $4$ are included, and no bound on the central multiplicity is assumed.
\end{theorem}

\subsection{The three further arithmetic families under GRH}
The following families are considered separately; GRH for one is not used to prove a statement about another.
\begin{enumerate}[label=(\roman*)]
\item Fix a primitive holomorphic cuspidal eigenform $f$ of even weight at least $2$, fixed level, and trivial nebentypus. Let $D$ range over all signed fundamental discriminants. The target is the primitive automorphic twist $L(s,f\otimes\chi_D)$, whose conductor $Q_D\asymp_f D^2$. The form $f$ may have complex multiplication.
\item Let $K$ range over non-Galois cubic fields of either fixed discriminant sign, without weights, and put $L_K(s)=\zeta_K(s)/\zeta(s)$. This is the entire degree-two standard Artin $L$-function of the $S_3$-closure, with conductor $Q_K=|D_K|$.
\item Let $K$ range over cyclic cubic fields, each once. If $\chi,\bar\chi$ are its two primitive cubic Dirichlet characters of conductor $q$, put $Z_K(s)=L(s,\chi)L(s,\bar\chi)=\zeta_K(s)/\zeta(s)$. The conductor and field discriminant are $Q_K=D_K=q^2$.
\end{enumerate}
In (i), the parameter is $|D|$; in (ii) and (iii), it is $|D_K|$. A dyadic family means $X/2<$ parameter $\le X$. Write $L_X=\log X$, $H=\log\log X$ and $H_Q=\log\log Q$. Constants below may depend on the fixed form in (i).

\begin{theorem}[Uniform full-count estimate under family GRH]\label{intro:quantile}
Assume GRH for the primitive $L$-functions of the selected family (i), (ii), or (iii). In (iii) this means GRH for the primitive cubic characters. Set $c=\cBM$ in (i) and (ii), and $c=1/(\pi\sqrt2)$ in (iii). For every fixed $0<\eps<1/16$ there are constants $A_\eps,C_\eps,X_\eps>0$ such that, for $X\ge X_\eps$ and every deterministic real $z$ with
\[
 H^{-1/2+\eps}\le z\le1,
\]
the original unweighted dyadic measure satisfies
\begin{equation}\label{intro:qbound}
 \Pbb_X\!\left\{\max_{C\in\{R^{\dist},R^{\mult},S\}}|C-cH_Q|>A_\eps zH_Q\right\}
 \le C_\eps\frac{\log H}{Hz^2}.
\end{equation}
The constants are independent of $z$. Central nonvanishing and simple zeros are not assumed.
\end{theorem}
The quantifier in \eqref{intro:qbound} allows a choice $z=z(X)$ in the indicated interval. It does not assert a single sample event for all $z$. It concerns the full count in probability, not its variance.

For fixed $\eta>0$, this gives an exceptional proportion $O_\eta(\log H/H)$ outside relative tolerance $\eta$. Two useful shrinking tolerances are
\begin{align}
 |C-cH_Q|&\le A H_Q^{2/3}(\log H_Q)^{1/3},
 &\text{exception }&O((\log H/H)^{1/3}),\label{intro:twothirds}\\
 |C-cH_Q|&\le A_\eps H_Q^{1/2+\eps},
 &\text{exception }&O_\eps((\log H)H^{-2\eps}).\label{intro:half}
\end{align}
These corollaries also hold for cumulative cutoffs: the exceptional counts are the displayed proportions times $X$ in (i),(ii) and times $\sqrt X$ in (iii). We do not assert an arbitrary-$z(X)$ cumulative estimate without specifying its dyadic envelope.

The coefficientwise twist $\sum_{n\ge1}\lambda_f(n)\chi_D(n)n^{-s}$ also satisfies the conclusions for (i). Its difference from the primitive twist is a fixed finite set of Euler factors. We retain the primitive conductor and the primitive GRH in this version; we do not assign the modified function a new primitive functional equation.

\subsection{Individual complex cubic derivatives}
The augmentation in (iii) is real. An individual cubic character instead gives a complex function on the real axis. A real zero of its derivative means a simultaneous zero of its real and imaginary parts; there is no associated sign-change statistic.
\begin{theorem}[Complex cubic characters under cubic-family GRH]\label{intro:complex}
Let $\mathcal C(X)$ contain all primitive exact-order-three Dirichlet characters with $X/2<q_\chi\le X$, including both conjugates. Assume GRH for this family. Let $R_\chi^{\mult}$ count the real zeros of $L'(s,\chi)$ on $[1/2,1]$ with complex-analytic multiplicity. For every fixed $0<\eps<1/16$,
\[
 \Pbb_{\mathcal C(X)}\{R_\chi^{\mult}>A_\eps(\log\log q_\chi)^{1/2+\eps}\}
 \ll_\eps \frac{\log H}{H^{1/2+\eps}}+H^{-2}\log H.
\]
In particular $R_\chi^{\dist}\le R_\chi^{\mult}=o(\log\log q_\chi)$ in density. The same proportion holds if each conjugate pair is counted once. The cumulative conductor and field-discriminant versions have the corresponding family cardinalities $\asymp X$ and $\asymp\sqrt X$.
\end{theorem}
This conclusion is an upper bound, not an asymptotic of the form ``$R\sim0$''. In the bulk used in the proof there is no real zero with probability $1-O(H^{-3})$; the entire closed interval need not be zero-free.

\subsection{The random theorem}
Fix a class of independent variables $X(p)\in\{0,1,-1\}$ with $\Pbb(X(p)=\pm1)=\theta_p/2$, where $0<\theta_*\le\theta\le1$ and $|\theta_p-\theta|\le A_*p^{-\eps_*}$. Define the nonvanishing random Euler product in $\Re s>1/2$ and
\[
 G_E(t)=-e^{-t}\frac{L_E'}{L_E}(1/2+e^{-t}).
\]
Let $N_E^\bullet(u,T)$ count its real zeros on $[u,u+T]$.
\begin{theorem}[Uniform random mean and exponential concentration]\label{intro:random}
Uniformly in the fixed class, deterministic $u\ge\log2$ and $T\ge1$,
\[
 \E N_E^{\bullet}(u,T)
 =\frac{e^{-\theta/2}}{\pi\sqrt2}T+O(e^{-5u/56}),\qquad \bullet\in\{\dist,\mult\}.
\]
For every fixed $\eta>0$ there are positive $c_\eta,C_\eta$, depending only on $\eta$ and the class, such that
\[
 \Pbb\left\{\left|N_E^\bullet(u,T)-\frac{e^{-\theta/2}}{\pi\sqrt2}T\right|>\eta T\right\}
 \le C_\eta e^{-c_\eta T},\qquad \bullet\in\{\dist,\mult\}.
\]
\end{theorem}

\subsection{What is averaged and where the proof changes scale}
The prime variables are independent in the model. Members of an arithmetic family are not independent random samples of an Euler product. The connection is a finite-polynomial orthogonality estimate with a power saving in the family parameter. We apply it to a finite Taylor polynomial of a characteristic function; its degree and its Euler cutoff are chosen so that their product is $o(\log X)$. This produces the required two-point and four-point comparisons without comparing exponentially many sign rectangles.

The variable $t=-\log(\sigma-1/2)$ turns a multiplicative interval of distances from the central point into an interval of fixed length. In the bulk, a fine mesh counts crossings, and Rolle's theorem controls the roots it misses. Near the central point a direct Gaussian comparison alone does not control poles of $L'/L$. There we combine the positive paired zero kernel, a meromorphic Jensen formula, and finitely many possible anchors. A single anchor whose value is not small suffices. Eight fixed anchors turn the bad event into an eight-dimensional small-ball event; independence of those anchors is neither true nor required.

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,>=Latex]
\draw[->,thick] (0,0)--(12.1,0) node[right] {$t$};
\draw[BMblue,very thick] (.4,.08)--(2.6,.08);
\draw[BMgreen,very thick] (2.6,.08)--(7.6,.08);
\draw[BMorange,very thick,dashed] (7.6,.08)--(10.5,.08);
\draw[BMgray,very thick,dotted] (10.5,.08)--(11.8,.08);
\foreach \x in {.4,2.6,7.6,10.5} \draw (\x,-.09)--(\x,.18);
\node[below] at (.4,-.12) {$\log2$};
\node[below] at (2.6,-.12) {$50\log H$};
\node[below,align=center] at (7.6,-.12) {$H-121\log H$};
\node[below,align=center] at (10.5,-.12) {$\log(L_X/\nu)$};
\node[above,align=center] at (1.5,.35) {initial part\\local moments};
\node[above,align=center] at (5.1,.35) {bulk\\crossing mesh};
\node[above,align=center] at (9.05,.35) {bridge\\eight anchors};
\node[above,align=center] at (11.25,.35) {central\\Jensen};
\end{tikzpicture}
\caption{The proof regions for the quantitative GRH theorem, with $L_X=\log X$ and $\nu=H^{1/2+\eps/4}$. This is a scale diagram, not a plot of numerical data or of region lengths. The central endpoint $\sigma=1/2$ corresponds to $t=+\infty$ and is treated in the $s$-plane.}
\label{intro:scales}
\end{figure}

For the unconditional families, the role of GRH is replaced by specifically proved density and mollified moment estimates. Their uniform ranges are kept separate. In particular, a power loss $X^\eps$ cannot be substituted for a logarithmic loss in a detector whose saving is only a power of $\log X$.

\input{figures/proof_map}

\subsection{Relation to earlier results and limitations}
Baker and Montgomery formulated the expected $\log\log$ order of the oscillation of quadratic $L$-function derivatives \cite{BakerMontgomery1990}. Their formulation includes a broader character convention; our main statement uses primitive characters indexed by fundamental discriminants. The Fekete-polynomial problem of Conrey, Granville, Poonen and Soundararajan \cite{CGPS2000} is related through character sums, but its unit-circle zero constant is not the Rice constant in \eqref{intro:constants}.

Related frameworks for using local splitting statistics in Artin-family
moment problems appear in Cho and Kim \cite{ChoKim2018}, and for
conductor-ordered cyclic extensions in Cho and Oh \cite{ChoOh2025}.
These works provided background for the extensions considered here.
The quantitative field and oriented-character inputs used in our proofs
are established separately in Section~\ref{families:section}.

\begin{table}[ht]
\centering\small
\begin{tabular}{>{\raggedright\arraybackslash}p{.19\textwidth}>{\raggedright\arraybackslash}p{.22\textwidth}>{\raggedright\arraybackslash}p{.23\textwidth}>{\raggedright\arraybackslash}p{.24\textwidth}}
\toprule
Work & Family and interval & Conclusion & Method or qualification\\
\midrule
Baker--Montgomery & quadratic characters; $(1/2,1)$ & conjectured $\log\log$ order of strict oscillation & character sums and logarithmic derivatives\\
Conrey--Granville--Poonen--Soundararajan & Fekete polynomials; unit circle and real intervals & zero-distribution results in a different normalization & finite character sums; no derivative Rice constant\\
Klurman--Lamzouri--Munsch & fundamental quadratic family; $(0,1)$ & almost-all lower bound of order $\log_2|D|/\log_4|D|$ & joint discrepancy; exception $O_A(Xe^{-(\log_3X)^A})$, fixed $A\ge1$\\
Lamzouri--Nath & $D=8m$; $[1/2,1]$ & almost-all upper bound $O(\log_2X\log_3X)$ & Jensen and zero density; exception $O(X\log_3X/\sqrt{\log_2X})$\\
This paper & the families of the theorems above; $[1/2,1]$ & exact density-one intensities; simultaneous three counts & finite-dimensional transfer, mesh excess, and central anchors; exception $o(\#\mathcal F)$ unconditionally, $O_\eta(\log H/H)$ under stated GRH\\
\bottomrule
\end{tabular}
\caption{Comparison of the strength and setting of the statements. The intervals and counting conventions are not identified across different works. The quadratic lower bound uses \cite[Theorem 1.8]{KLM2024}, version 2; the upper bound uses \cite[Theorem 1.2]{LamzouriNath2025}, version 2.}
\end{table}

None of the arithmetic statements here asserts a central nonvanishing theorem, a central simplicity theorem, a full-count central limit theorem, or a family mean asymptotic. The quantitative conductor thresholds are not made effective. The constants arising from fixed-anchor covariance matrices can be very large. Actual low-conductor computations therefore test local identities and sign brackets, not the asymptotic Rice constant. Removing GRH for the three further families requires new estimates in the central mesoscopic range; currently available estimates with conductor-power losses do not supply those inputs by direct substitution.