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\title[The positive directions of the archimedean place]{Cross-Energy of Dirichlet Partial Sums.\\ Part 16: How many positive directions does the archimedean place have? A counting law with a boundary phase}
\author{Pedro Caceres}
\address{Universidad Europea de Valencia, Valencia, Spain}
\email{pedrojesus.caceres@universidadeuropea.es}
\subjclass[2020]{Primary 11M26; Secondary 11M06, 47A75, 35P20, 47B35}
\keywords{Weil's quadratic form, archimedean place, Riemann--Siegel theta function, Dirichlet $L$-functions, complete Bernstein functions, eigenvalue asymptotics on an interval, Szeg\H{o} limit theorems}
\date{November 2026 (note, v1.0); renumbered 2 November 2026; v1.1, 10 November 2026 (the law stated as a conjecture, the smooth constant identified with Widom's second term) (formerly a standalone note). Part~16 of the book-series \emph{Cross-Energy of Dirichlet Partial Sums} (forty-seven parts, Parts 0--46; block III: The margin and the constant $\widehat B$). Part~0 \cite{CE0} is the guide. In book version~1 (2 November 2026) this was Part~15; the present numbering is that of book version~2 (10 November 2026).}
\begin{document}

\begin{abstract}
Restrict the archimedean part of Weil's quadratic form, for $\zeta$ or for a primitive real Dirichlet $L$-function of conductor $q$ and parity $\kappa$, to test functions supported in an interval of length $a$, and let $n_+(a)$ be the number of its positive directions, the directions that the pole and the primes must absorb. A Szeg\H{o}-type argument gives $n_+(a)=t_0a/\pi+o(a)$, where $t_0$ is the zero of $\vartheta'$. We observe that the form is $\Lambda-\psi_\kappa(-\Delta)$ with $\psi_\kappa$ a complete Bernstein function (proved) and, using the boundary phase of Kaleta, Kwa\'snicki and Ma{\l}ecki, we state, as a conjecture, the asymptotic law
\[
n_+(a)=\Big\lfloor\frac{t_0a}\pi+\frac{2\varpi_\kappa(t_0)}\pi\Big\rfloor ,
\]
with an explicit phase $\varpi_\kappa$ that depends only on the parity, the conductor entering only through $t_0$. We test it in seven cases, $\zeta$ and $\chi_{-3},\chi_{-4},\chi_5,\chi_{-7},\chi_8,\chi_{-8}$. On the finest mesh the shifts of the jumps $m=2,\dots,12$ bracket the prediction $2\varpi_\kappa(t_0)/\pi$, which ranges from $0.18$ to $0.59$, to within $1$ to $3\cdot10^{-3}$ for the characters and $3\cdot10^{-2}$ for $\zeta$. A second, independent implementation extrapolates in the mesh on three to nine jumps per case; the extrapolated shifts lie above the prediction by $0.2$ to $2.3\cdot10^{-4}$ in every case, a one-sided offset of the size of the extrapolation error, which we have checked to be a bias in one case only. The first jump is an exception, and for $\zeta$ the shift reaches its limit only after about ten jumps. A smoothed count removes the integer part and leaves ``degree plus constant'' with the boundary phase as constant; on a circle the constant is carried by the end points and cancels between an arc and its complement; and treating $-\log q$ as a degree, $q\leftrightarrow1/q$ acts on the band edge as $t_0\mapsto4\pi^2/t_0$ to leading order, with $\zeta$ at the self-dual point $t_0\approx2\pi$. We record what the law implies for the signature of Weil's form and where it stands with respect to the Riemann--Roch formulas for $\overline{\operatorname{Spec}\mathbb Z}$. The law is a conjecture supported by these computations. The Bernstein structure is proved here; the first-order count is quoted from Part~14 and its proof is not repeated.
\end{abstract}

\maketitle

\section{The question}
For $\zeta$ the explicit formula writes Weil's quadratic form, on real test functions $g$, as
\begin{gather*}
Q(g)=W(g)-E(g),\qquad W(g)=2\widehat g(\tfrac i2)\widehat g(-\tfrac i2),\\
E(g)=A(g)+\sum_n\frac{\Lambda(n)}{\sqrt n}\int g(x)\big(g(x+\log n)+g(x-\log n)\big)\,dx,
\end{gather*}
with $\widehat g(t)=\int g(x)e^{-itx}dx$ and the archimedean part
\[
A(g)=-\frac1\pi\int_{-\infty}^{\infty}|\widehat g(t)|^2\,\vartheta'(t)\,dt,\qquad \vartheta'(t)=\tfrac12\big(\re\psi(\tfrac14+\tfrac{it}2)-\log\pi\big),
\]
$\vartheta$ the Riemann--Siegel theta function \cite{Bo,CC}. The pole $W$ is a hyperbolic plane, with one positive and one negative direction, and RH is $Q\ge0$ \cite{We,Bo}. Following Weil's proof for curves one reads $W$ as the product of two degrees and $E$ as a self-intersection: RH implies that $E$ has at most one positive direction on every space of test functions (Lemma~\ref{lem:inertia}), as the Hodge index theorem forces for curves \cite{CE14}. The archimedean place alone has many positive directions, and the primes must remove all of them but the pole's. This note is about how many there are.

For a primitive real character $\chi$ modulo $q$ of parity $\kappa\in\{0,1\}$ the completed function $(q/\pi)^{(s+\kappa)/2}\Gamma(\frac{s+\kappa}2)L(s,\chi)$ has no pole, and the archimedean part is the same with
\[
\vartheta_\chi'(t)=\tfrac12\big(\re\psi(\tfrac14+\tfrac\kappa2+\tfrac{it}2)+\log\tfrac q\pi\big);
\]
$\zeta$ is $\kappa=0$, $q=1$. For $a>0$ let $n_+(a)$ be the largest dimension of a subspace of real $g\in H^1_0([0,a])$ on which $A$ is positive definite.

\begin{theorem}[first order \cite{CE14}]\label{thm:first}
If $\vartheta_\chi'(0)<0$, let $t_0$ be the positive zero of $\vartheta_\chi'$. Then $n_+(a)=t_0a/\pi+o(a)$. If $\vartheta_\chi'(0)\ge0$, $n_+(a)=0$ for all $a$.
\end{theorem}
The symbol $\sigma=-\vartheta_\chi'/\pi$ is even, strictly decreasing in $|t|$, positive exactly on $|t|<t_0$ and $\sim-\frac1{2\pi}\log|t|$, unbounded below, so Landau's eigenvalue distribution theorem \cite{La} does not apply directly. The proof in \cite{CE14} truncates it: for the upper bound, $A\le2\pi W_a(\max(\sigma,-M))$ and Landau's theorem applies to the bounded symbol $\max(\sigma,-M)+M$ with compact support, counting eigenvalues above $M$; for the lower bound, the prolate spheroidal functions of the band $|t|<c$, $c<t_0$, cut off smoothly to the window, span a space of dimension $c\,a/\pi+O(\log a)$ \cite{LW} on which $A$ is positive. If $\vartheta_\chi'(0)\ge0$ then $\sigma<0$ for $t\ne0$ and $A$ is negative definite. The theorem is imported: this part does not prove it. In particular the lower bound needs the positivity of $A$ on the cut-off prolate space in spite of the unbounded negative tail of $\sigma$, that is, a bound for the energy that the cut-off functions leave outside the band; that estimate is in \cite{CE14} and is not re-derived or re-checked here. Nothing below depends on more than the leading term $t_0a/\pi$. For $\zeta$, $t_0=6.289836$, so $n_+\approx2.0021\log N$ in the window $[0,\log N]$. The condition $\vartheta_\chi'(0)<0$ holds for odd $\chi$ with $q<\pi e^{-\psi(3/4)}=9.31$ and even $\chi$ with $q<\pi e^{-\psi(1/4)}=215.3$.

\paragraph{Place in the book.} This is Part~16 of the series, the last chapter of block III (The margin and the constant $\widehat B$). It builds on Part~14 \cite{CE14}, where the margin is written as $W-E$, the first-order count $t_0a/\pi$ of the positive directions of the archimedean place is proved, and the law below is first recorded; this part states that law with its boundary phase and tests it for $\zeta$ and six real Dirichlet $L$-functions. No later part cites it.

\section{The structure of the symbol}
\begin{lemma}[complete Bernstein structure]\label{lem:cbf}
Put $c=\frac14+\frac\kappa2$ and $\psi_\kappa(\xi)=\vartheta_\chi'(\sqrt\xi)-\vartheta_\chi'(0)$, $\xi\ge0$. Then
\[
\psi_\kappa(\xi)=\frac12\sum_{n\ge0}\frac1{n+c}\;\frac{\xi}{\xi+4(n+c)^2},
\]
a complete Bernstein function, independent of $q$, with $\psi_\kappa(\xi)=\frac14\log\xi+O(1)$. Hence on the window
\[
A(g)=2\big\langle(\Lambda-\psi_\kappa(-\Delta))g,g\big\rangle,\qquad\Lambda=-\vartheta_\chi'(0)=\psi_\kappa(t_0^2),
\]
with $\langle\psi(-\Delta)g,g\rangle=\frac1{2\pi}\int|\widehat g(t)|^2\psi(t^2)\,dt$, and $n_+(a)$ is the number of eigenvalues of $\psi_\kappa(-\Delta)$ on an interval of length $a$, with zero exterior condition, strictly below the level $\Lambda$ ($H^1_0$ is a form core, so the counts agree).
\end{lemma}

\begin{proof}
From $\psi(z)=-\gamma_E+\sum_{n\ge0}(\frac1{n+1}-\frac1{n+z})$ and $\re\frac1{n+c+it/2}=\frac{n+c}{(n+c)^2+t^2/4}$,
$\re\psi(c+\frac{it}2)-\psi(c)=\sum_n\big(\frac1{n+c}-\frac{n+c}{(n+c)^2+t^2/4}\big)=\sum_n\frac1{n+c}\frac{t^2}{t^2+4(n+c)^2}$. Each $\xi\mapsto\xi/(\xi+z)$, $z>0$, is complete Bernstein, the weights are positive and $\sum_n(n+c)^{-3}<\infty$. The rest is Plancherel.
\end{proof}

So the conductor does not change the operator, only the level: $\zeta$, $\chi_5$, $\chi_8$ share $\psi_0$, and $\chi_{-3},\chi_{-4},\chi_{-7},\chi_{-8}$ share $\psi_1$.

\section{The law}
For $\psi(-\Delta)$ on an interval, $\psi$ complete Bernstein, Kwa\'snicki found the generalized eigenfunctions on the half-line, $\sin(\mu x+\varpi_\mu)-G_\mu(x)$ with $G_\mu$ completely monotone \cite{Kw1}, and the eigenvalues of the fractional Laplacian on an interval \cite{Kw2}; Kaleta, Kwa\'snicki and Ma{\l}ecki proved, when $\xi\psi'(\xi)\to\infty$, that on an interval of length $a$ \cite{KKM}
\[
\lambda_n=\psi(\mu_n^2)+O(\tfrac1n),\qquad\mu_na=n\pi-2\varpi_{\mu_n},\qquad
\varpi_\mu=\frac1\pi\int_0^\infty\frac{\mu}{r^2-\mu^2}\log\frac{\psi'(\mu^2)(\mu^2-r^2)}{\psi(\mu^2)-\psi(r^2)}\,dr\in[0,\tfrac\pi2).
\]
For $\psi(\xi)=\xi^{\alpha/2}$, $\varpi_\mu=(2-\alpha)\pi/8$. (We write $\varpi$ for this boundary phase, which \cite{Kw1,KKM} call $\vartheta_\mu$, to keep $\vartheta$ for the Riemann--Siegel theta function.) Our $\psi_\kappa$ grows like a logarithm, $\xi\psi_\kappa'(\xi)\to\frac14$, just outside that hypothesis, which excludes $\log(1+\xi)$ explicitly. Our question is at a fixed level on a growing interval, and there an error $O(1/n)$ in $\lambda_n$ is an error $O(1/a)$ in $\mu_n$, that is $O(1)$ in $t_0a/\pi$, the size of the shift to be predicted. So even a transfer of \cite{KKM} would give only $n_+(a)=t_0a/\pi+O(1)$; the law below needs the eigenvalue formula at the level $\Lambda=\psi_\kappa(t_0^2)$ with error $o(1/a)$, smaller than the gap between eigenvalues, which is stronger than what \cite{KKM} prove under their hypothesis. If it holds, the eigenvalues below the level are those with $\mu_n<t_0$:

\begin{conjecture}[the counting law; asymptotic]\label{conj:law}
Let $\varpi_\kappa(\mu)$ be the phase of $\psi_\kappa$. Then $n_+(a)=\lfloor t_0a/\pi+2\varpi_\kappa(t_0)/\pi\rfloor$ for $a$ outside neighbourhoods of the jumps that shrink as $a\to\infty$. Equivalently, the count reaches $m$ at $a_m$ with $t_0a_m/\pi=m-\delta_m$ and $\delta_m\to\delta=2\varpi_\kappa(t_0)/\pi$ as $m\to\infty$.
\end{conjecture}

The phase is a universal function of the level for each parity (Figure~\ref{fig:rr}(a)). Both phases vanish at $\mu=0$ and tend to $\pi/4$ as $\mu\to\infty$, the logarithmic case, the limit $\alpha\to0$ of $(2-\alpha)\pi/8$. The odd phase increases monotonically to $\pi/4$; the even phase rises above it, to a maximum $2\varpi/\pi\approx0.59$ near $\mu\approx1.2$, and then decreases to $\pi/4$. We computed it with two quadratures (adaptive, and a trapezoid rule in $\log r$ with $4\cdot10^6$ nodes), which agree to better than $10^{-6}$ in $2\varpi/\pi$, and checked the routine on $\xi^{\alpha/2}$, $\alpha=2,1,\frac12$.

\section{The test}
\emph{Method.} With the $k-1$ interior hat functions of step $h$ on $[0,a]$, $a=kh$, the Gram matrix of $A$ is the Toeplitz matrix $T_{k-1}(f_h)$ of the symbol
\[
f_h(v)=\sum_{j\in\mathbb Z}\sigma\Big(\frac{v+2\pi j}h\Big)\,\mathrm{sinc}^4\Big(\frac{v+2\pi j}2\Big),\qquad\sigma=-\vartheta_\chi'/\pi,
\]
up to a positive factor ($\mathrm{sinc}\,x=\sin x/x$), so by Sylvester's law the number of positive directions on the hat space is the number of positive pivots of $T_{k-1}(f_h)$, and the Levinson recursion gives the pivots of all sections at once. The hat space is a subspace of $H^1_0$, so the discrete count is a lower bound for $n_+(a)$, its jumps come late, and the measured $\delta$ increases to its limit as $h\to0$. The Toeplitz entries were computed by FFT on $2^{12}$ to $2^{18}$ points with $241$ aliasing terms; the Levinson counts agree with full eigenvalue counts in all cases checked ($n$ up to $4000$; smallest $|$eigenvalue$|$ $\ge3.7\cdot10^{-3}$).

\emph{Result.} For each case and $h=0.02,0.01,0.005,0.0025$ we located the jumps $m=1,\dots,12$; a jump is located to $t_0h/\pi$ in $\delta$. With $h=0.0025$:
\begin{center}\small
\begin{tabular}{lcccccc}
\toprule
 & $\kappa$ & $q$ & $\Lambda=-\vartheta_\chi'(0)$ & $t_0$ & predicted $2\varpi_\kappa(t_0)/\pi$ & measured $\delta$, $m=2,\dots,12$\\
\midrule
$\zeta$ & $0$ & $1$ & $2.686092$ & $6.289836$ & $0.50902$ & $0.483$ to $0.522$; $0.5089$ (extrapolated)\\
$\chi_{-3}$ & $1$ & $3$ & $0.565989$ & $2.106205$ & $0.44053$ & $0.4379$ to $0.4411$\\
$\chi_{-4}$ & $1$ & $4$ & $0.422148$ & $1.563882$ & $0.39707$ & $0.3953$ to $0.3976$\\
$\chi_{5}$ & $0$ & $5$ & $1.881373$ & $1.349504$ & $0.58649$ & $0.5846$ to $0.5869$\\
$\chi_{-7}$ & $1$ & $7$ & $0.142340$ & $0.719570$ & $0.24007$ & $0.2392$ to $0.2403$\\
$\chi_{8}$ & $0$ & $8$ & $1.646371$ & $0.965814$ & $0.58471$ & $0.5835$ to $0.5850$\\
$\chi_{-8}$ & $1$ & $8$ & $0.075575$ & $0.500953$ & $0.17551$ & $0.1750$ to $0.1756$\\
\bottomrule
\end{tabular}
\end{center}
The ranges are the union over the eleven jumps of the brackets; for $\zeta$, whose $t_0$ is largest, the bracket is $0.005$ wide and the range is wider; bisection in $a$ for the tenth jump gives $0.4742$, $0.4904$, $0.4991$, $0.5037$ for $h=0.04$, $0.02$, $0.01$, $0.005$, with geometric extrapolation $0.5089$. For the characters the shift does not depend on $m$ for $m\ge2$; for $\zeta$ it oscillates and settles (see below). In every case it converges from below as $h$ decreases; for $\chi_5$ the brackets at the tenth jump are $[0.575,0.584]$, $[0.584,0.588]$, $[0.584,0.586]$, $[0.585,0.586]$ for $h=0.02$ to $0.0025$. A second discretization, by the sine basis $\sin(j\pi x/a)$, gives for the sixth jump of $\chi_{-4}$ the lower bounds $0.363$, $0.378$, $0.384$ with $4a$, $8a$, $12a$ functions, converging more slowly, from below, towards $0.397$.

\emph{Independent check.} The computation was re-implemented from scratch, with a different derivation of the Gram matrix: in $x$-space, from Gauss's integral for $\psi$, $A(g)=-\big[\int_0^\infty\big(\frac{e^{-u}}uR(0)-\frac{e^{-cu}}{1-e^{-u}}R(\frac u2)\big)du+\log\frac q\pi\,R(0)\big]$ with $R$ the autocorrelation of $g$ (for hats, $h$ times the cubic B-spline), its own trigamma for $\vartheta''$, the jumps located as zeros of the $m$-th eigenvalue in $a$, meshes $ht_0=0.12$, $0.06$, $0.03$, $0.015$ (and $0.0075$ for some jumps), and Aitken extrapolation (first-order convergence, ratio $1.92$--$1.95$). Its Toeplitz entries agree with a direct $t$-integral to $10^{-9}$ and its phases with ours to better than $10^{-6}$. Extrapolated shifts against the prediction:
\begin{center}\small
\begin{tabular}{lcccc}
\toprule
 & predicted & extrapolated $\delta_m$ & jumps & first jump $\delta_1$\\
\midrule
$\zeta$ & $0.509024$ & $0.50905$ to $0.50921$ & $m=13,14,20$ & $0.571$\\
$\chi_{-3}$ & $0.440532$ & $0.44062$ to $0.44066$ & $m=2,3,5,8,10$ & $0.4449$\\
$\chi_{-4}$ & $0.397071$ & $0.39709$ to $0.39721$ & $m=2,\dots,10$ & $0.3984$\\
$\chi_{5}$ & $0.586486$ & $0.58651$ to $0.58660$ & $m=3,\dots,10$ & $0.6086$\\
$\chi_{-7}$ & $0.240069$ & $0.24019$ to $0.24024$ & $m=2,3,5,8,10$ & $0.2401$\\
$\chi_{8}$ & $0.584709$ & $0.58473$ to $0.58479$ & $m=2,3,5,8,10$ & $0.5905$\\
$\chi_{-8}$ & $0.175515$ & $0.17569$ to $0.17574$ & $m=2,3,5$ & $0.1757$\\
\bottomrule
\end{tabular}
\end{center}
All the extrapolated values lie above the prediction, by $0.2$ to $2.3\cdot10^{-4}$; we attribute this to a bias of the Aitken extrapolation, checked in one case: for $\chi_{-4}$ a finer mesh moves it to $0.39709$. Two departures. The first jump does not follow the law in five cases, always with $\delta_1$ above the limit (by $0.062$ for $\zeta$, $0.022$ for $\chi_5$, $0.006$ for $\chi_8$, $0.004$ for $\chi_{-3}$, $0.001$ for $\chi_{-4}$), so the first positive direction appears earlier than the law says. For $\zeta$ the shift alternates around its limit before settling, $\delta_m-\delta=-0.019$, $+0.012$, $-0.007$, $+0.005$, $-0.003$ for $m=2,\dots,6$, $-0.0009$ at $m=8$, $-0.0003$ at $m=10$ and below $2\cdot10^{-4}$ for $m=13$ to $20$, an even--odd effect; for the characters, whose $t_0$ is smaller, it is flat from $m=2$ on. A plausible reason, not tested, is that for $\zeta$ the first jumps occur in windows that are short compared with the decay length of the kernel in $x$. None of this changes the integer count for $m\ge2$, since the deviations are far below the distance between jumps.

\begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{CE_rr.png}
\caption{(a) The phases $2\varpi_\kappa(\mu)/\pi$ of $\psi_0$ (even) and $\psi_1$ (odd) as functions of the level, and the measured shifts at $\mu=t_0(\chi)$. (b) Predicted and measured shifts in the seven cases (hats of step $0.0025$, jumps $2$ to $12$).}\label{fig:rr}
\end{figure}


\section{A smooth count}
The integer part in Conjecture~\ref{conj:law} disappears if the count is smoothed. For a smooth $F$ with $F\to0$ below the level and $F\to1$ above, put
\[
h^0_F(a)=\operatorname{Tr}F\big(W_a(\sigma)\big),\qquad \sigma=-\vartheta_\chi'/\pi ,
\]
$W_a(\sigma)$ the operator on $L^2(0,a)$ with $\langle W_a(\sigma)g,g\rangle=\frac1{2\pi}\int|\widehat g|^2\sigma$, so that $A(g)=2\pi\langle W_a(\sigma)g,g\rangle$. If the eigenvalues near the level are $\sigma(\mu_n)$ with $\mu_na=n\pi-2\varpi_\kappa(\mu_n)$, the Euler--Maclaurin formula, with $\varpi_\kappa(0)=0$, gives
\begin{equation}\label{eq:smooth}
h^0_F(a)=\frac a\pi\int_0^\infty F(\sigma(\mu))\,d\mu+\chi_F+o(1),\qquad
\chi_F=\frac2\pi\int_0^\infty F(\sigma(\mu))\,d\varpi_\kappa(\mu)-\frac12F(\sigma(0)),
\end{equation}
and $\chi_F\to2\varpi_\kappa(t_0)/\pi-\frac12$ as $F$ tends to the step at $0$, the mean of $\lfloor x+\delta\rfloor-x$. This is a ``degree plus constant'' formula with real-valued dimension, of the shape of the van der Geer--Schoof $h^0$ \cite{GS}, and its constant is the boundary phase. For $F(x)=\frac12(1+\tanh(x/\varepsilon))$, for which $F(\sigma(\mu))\sim\mu^{-1/(\pi\varepsilon)}$ and the trace converges if $\varepsilon<1/\pi$, we computed $h^0_F(a)$ from all the generalized eigenvalues (with respect to the mass matrix) of the hat discretization (\texttt{traza\_suave.py}, \texttt{traza\_pocos.py}; $\chi_F$ by integration by parts, \texttt{chi\_pred.py}):
\begin{center}\small
\begin{tabular}{lcccccc}
\toprule
 & $\varepsilon$ & $h_1$ & $h_1/2$ & $h_1/4$ & extrapolated & $\chi_F$ in \eqref{eq:smooth}\\
\midrule
$\chi_{-7}$, $h_1=0.02$ & $0.02$ & $-0.263096$ & $-0.261988$ & $-0.261420$ & $-0.260823$ & $-0.260834$\\
$\zeta$, $h_1=0.01$ & $0.05$ & $0.000011$ & $0.004776$ & $0.007266$ & $0.00999$ & $0.009889$\\
\bottomrule
\end{tabular}
\end{center}
The difference $h^0_F(a)-\frac a\pi\int F(\sigma)$ is constant in $a$ to six digits from $a=25$ for $\chi_{-7}$ and from $a=10$ for $\zeta$; the mesh error is first order, and the geometric extrapolation agrees with \eqref{eq:smooth} to $1.1\cdot10^{-5}$ and $10^{-4}$. The sharp limits are $\chi=-0.25993$ for $\chi_{-7}$ and $+0.00902$ for $\zeta$. Status: \eqref{eq:smooth} is numerical; for bounded symbols, two-term formulas of this kind are theorems of Widom and Sobolev \cite{Wi,So}, whose hypotheses exclude our logarithmic symbol.

\emph{Identification with Widom's second term.} For a smooth $F$ and a suitable bounded symbol (with $F\circ\sigma-F(0)$ integrable, which makes the trace finite), Widom's trace formula \cite{Wi} reads
\[
\operatorname{Tr}F\big(W_a(\sigma)\big)=\frac a{2\pi}\int F(\sigma(\xi))\,d\xi+\mathcal B(\sigma;F)+o(1),\qquad
\mathcal B(\sigma;F)=\frac1{4\pi^2}\iint\frac{U\big(\sigma(\xi_1),\sigma(\xi_2);F\big)}{(\xi_1-\xi_2)^2}\,d\xi_1d\xi_2,
\]
with $U(x,y;F)=\int_0^1\frac{F((1-t)x+ty)-(1-t)F(x)-tF(y)}{t(1-t)}\,dt$ (one half of $\mathcal B$ for each end of the interval). The constant $\chi_F$ of \eqref{eq:smooth} is this second term for our symbol: a direct evaluation of the double integral (\texttt{widom\_check.py}, written by the referee of this part and rerun by us; stable to $10^{-8}$ under changes of the quadrature) gives $\mathcal B(\sigma;F)=0.00988902$ for $\zeta$ ($\varepsilon=0.05$) and $-0.26083400$ for $\chi_{-7}$ ($\varepsilon=0.02$), against $\chi_F=0.009889$ and $-0.260834$ in the table. As a check of the constant $\frac1{4\pi^2}$: for $F(x)=x^2$ one has $U=-(x-y)^2$, and the formula reduces to $\operatorname{Tr}W_a^2=a\int|k|^2-\int|u||k(u)|^2du$, which is elementary. So \eqref{eq:smooth} is Widom's formula extended, numerically, to a symbol of logarithmic growth, and the two expressions for the constant, one through the boundary phase $\varpi_\kappa$ of \cite{KKM} and one through $\mathcal B$, agree. In the step limit this gives an expression for the constant of Conjecture~\ref{conj:law} that does not use the phase; we have not tested that limit. No proof of the identity $\chi_F=\mathcal B(\sigma;F)$ for complete Bernstein symbols is given here.


\section{Complements on a circle}
A first candidate for a dual count is the complement. Put the window on a circle of length $L$ and let $W^{S^1}$ be the archimedean operator there, $\langle W^{S^1}g,g\rangle=\frac1L\sum_k\sigma(\omega_k)|\widehat g(\omega_k)|^2$, $\omega_k=2\pi k/L$, whose eigenvalues are exactly the $\sigma(\omega_k)$. Restrict it to an arc $I$ of length $a$ and to the complementary arc $I^c$ of length $L-a$, and write $\deg I=\frac{|I|}\pi\int_0^\infty F(\sigma(\mu))\,d\mu$ for the volume term of \eqref{eq:smooth}. Numerically (\texttt{circulo.py}, hats of step $h$, the circle operator folded exactly):
\begin{center}\small
\begin{tabular}{lcccc}
\toprule
 & $L$, $h$, $\varepsilon$ & $a$ & $h^0_F(I)+h^0_F(I^c)-h^0_F(S^1)$ & $h^0_F(I)-h^0_F(I^c)-(\deg I-\deg I^c)$\\
\midrule
$\chi_{-7}$ & $70$, $0.02$, $0.02$ & $25$, $30$, $35$ & $-0.526190$, $-0.526192$, $-0.526192$ & $2\cdot10^{-6}$, $0$, $0$\\
$\chi_{5}$ & $60$, $0.02$, $0.05$ & $20$, $27$ & $+0.151729$, $+0.151729$ & $0$, $0$\\
$\zeta$ & $30$, $0.01$, $0.05$ & $10$, $13$ & $+0.000015$, $+0.000016$ & $7\cdot10^{-6}$, $3\cdot10^{-6}$\\
\bottomrule
\end{tabular}
\end{center}
and $h^0_F(S^1)=\deg S^1$ to the six digits printed in every case. So, to this accuracy, (i) the closed circle has no constant; (ii) the difference between an arc and its complement has no constant, $h^0_F(I)-h^0_F(I^c)=\deg I-\deg I^c$; and (iii) the sum carries $2\chi_F$, which is $-0.526192=2\times(-0.263096)$ for $\chi_{-7}$, twice the value of the interval at the same mesh, $0.151729\approx2\chi_F$ for $\chi_5$ (at this mesh) and $\approx0$ for $\zeta$ (the mesh offset of $\zeta$ at $h=0.01$ cancels its $\chi_F$). The constant is local: each of the four end points contributes $\chi_F/2$, which tends to $\varpi_\kappa(t_0)/\pi-\frac14$ as $F$ tends to the step. With $K=S^1$ and $K-D=I^c$, (ii) reads $h^0(D)-h^0(K-D)=\deg D-\deg(K-D)$, a duality that removes the boundary constant. But it is not Serre duality: $h^0(K-D)$ does not vanish for large $D$, the circle is not canonical ($L$ is free), and (i)--(iii) are those of an Euler characteristic with boundary, of Mayer--Vietoris type, with the boundary phase as a local index at each end point. Status: numerical.


\section{The conductor as a degree}
Lemma~\ref{lem:cbf} says that the conductor only moves the level: $\Lambda=-\frac12(\psi(c)+\log\frac q\pi)$. Treat $d=-\log q$ as a degree, letting $q>0$ be real ($q<1$ is formal), so that $\Lambda(d)=\Lambda(0)+\frac d2$, and consider the reflection $d\mapsto-d$, $q\mapsto1/q$, whose fixed point is $q=1$. Since $\vartheta_q'(t)=\frac12\log\frac{qt}{2\pi}+O(t^{-2})$ (Stirling), the band edge is $t_0(q)\approx2\pi/q$ and the reflection acts as
\[
t_0\longmapsto\frac{4\pi^2}{t_0},
\]
the inversion $t\mapsto1/t$ of the theta function, rescaled by $2\pi$, that gives van der Geer and Schoof their Riemann--Roch formula \cite{GS}. Its fixed point is $t_0=2\pi$, and $\zeta$ sits there: $t_0(\zeta)=6.289836$ against $2\pi=6.283185$. This is why the slope $t_0/\pi=2.0021$ of $\zeta$ is so close to $2$: $\zeta$ is the self-dual point, and $0.0021$ is the Stirling correction. With $n_+\approx t_0a/\pi$, the logarithm of the count then satisfies, to leading order,
\[
\log n_+(a;d)-\log n_+(a;-d)\approx2d,
\]
a ``degree'' relation of the shape of $h^0(D)-h^0(K-D)=\deg D$ for the logarithmic $h^0$ of \cite{GS}, with $n_+=0$ once $d$ is so negative that $\Lambda\le0$, as $h^0(D)=0$ in negative degree. But the duality is not exact (\texttt{dualidad\_q.py}):
\begin{center}\small
\begin{tabular}{lcccccc}
\toprule
$d=-\log q$ & $0.25$ & $0.5$ & $1$ & $1.5$ & $2$ & $3$\\
\midrule
$t_0(d)\,t_0(-d)/4\pi^2$, $\kappa=0$ & $1.0024$ & $1.0033$ & $1.0102$ & $1.0531$ & $1.1955$ & $1.8759$\\
$t_0(d)\,t_0(-d)/4\pi^2$, $\kappa=1$ & $1.0024$ & $1.0033$ & $1.0060$ & $0.9837$ & $0.7474$ & ---\\
\bottomrule
\end{tabular}
\end{center}
It holds while both band edges are in the logarithmic range of $\vartheta'$ and fails beyond, where the full gamma factor departs from Stirling. The exact inversion lives one level up, in the Gaussian $e^{-\pi qx^2}$ and its Fourier transform, whose Mellin transform is the gamma factor; the count sees it only asymptotically. Status: elementary (Stirling) plus numerical; the real conductors are $q\ge1$, so the dual side $q<1$ is a formal extension.

\section{What the law says about Weil's form}
\begin{lemma}[inertia \cite{CE14}]\label{lem:inertia}
On any space of test functions, if $n_-(Q)$ and $n_+(E)$ count the negative directions of $Q$ and the positive directions of $E=W-Q$, then $n_-(Q)-1\le n_+(E)\le n_-(Q)+1$, and $n_+(E)=n_-(Q)$ on the hyperplane $\{\widehat g(-\frac i2)=0\}$, where $W$ vanishes.
\end{lemma}
This is subadditivity of $n_+$ with $n_+(\pm W)\le1$. Applied to the form $Q_R=W-A$ of the pole and the archimedean place without primes, it gives $n_-(Q_R)\ge n_+(A)-1$: the form without primes must be indefinite as soon as $A$ has two positive directions. For $\zeta$ this happens at $\log N\approx0.754$, $N\approx2.13$ (the second jump), and numerically $Q_R$ loses positivity at $\log N=0.744$, $N=2.106$ \cite{CE14}; Yoshida proved positivity below $N=2$ \cite{Yo}. With all the primes, $E$ has one positive direction in every window we computed, up to $N=1097$ \cite{CE14}, as RH requires; so, numerically, the orbits of the prime powers turn $n_+(A)$ positive directions into one, that is $n_+(A)-1$, which by Conjecture~\ref{conj:law} is $\lfloor t_0\log N/\pi+0.509\rfloor-1$ for $\log N$ beyond the first jumps; in \cite{CE14} they do so only all together and only with the prime powers (numerical). For the Davenport--Heilbronn function the same count holds for the archimedean place, and, numerically, $E$ gains one pair of positive directions for each off-line zero that the window resolves \cite{CE14}; Bombieri proved that if finitely many zeros are off the line, large truncations of Weil's form have half as many negative eigenvalues as off-line zeros \cite{Bo}, and the pairs are the window form of this count.

\section{Relation with Riemann--Roch}
The law has the shape ``dimension $=$ degree $\times$ slope $+$ constant''. It is not one of the known Riemann--Roch formulas for $\overline{\operatorname{Spec}\mathbb Z}$. Connes and Consani's integer-valued dimension of $H^0$ of the divisor $\log N\{\infty\}$ is $\lceil\log(2\lfloor N\rfloor+1)/\log3\rceil\approx0.91\log N$ \cite{CoCo1,CoCo2}; the log-theta number of van der Geer and Schoof grows like $\log N$ \cite{GS}. Both measure the integers in $[-N,N]$, the first by the dimension over $\mathbb S[\pm1]$ of a minimal set of generators, the second by a Gaussian-weighted theta sum, $\log\sum_ne^{-\pi n^2/N^2}=\log N+o(1)$; our count is spectral, the number of degrees of freedom of the band $|t|<t_0$ in a window of length $\log N$. It is also not the spectral count of the prolate operator, which fixes the window ($\lambda=\sqrt2$) and counts by energy, reproducing the Riemann--von Mangoldt count of the zeros \cite{CM,CCM}. A natural reading, which is a heuristic analogy and not a theorem, is the counting formula $N(T)=\vartheta(T)/\pi+1+S(T)$: the band $|t|<t_0$ is where $\vartheta$ decreases, the archimedean surplus counts its degrees of freedom, and the $1$ is the pole, the one direction that survives. The smooth count \eqref{eq:smooth} removes the integer part and has real-valued dimension, like the $h^0$ of van der Geer and Schoof, whose Riemann--Roch formula is the functional equation of the theta function, the same theta whose Mellin transform is the gamma factor behind $\vartheta'$. What it still lacks to be a Riemann--Roch formula in the proper sense is a second term and a duality, $h^0-h^1$ with $h^1(D)=h^0(K-D)$: we have only one count, and the negative directions of $A$ are infinitely many, so $h^1$ cannot be their number. The slope $t_0/\pi=2.0021$ is not an integer, and the constant comes from the boundary of the window, through $\varpi_\kappa$.

\section{Open questions}
(1) Prove $n_+(a)=t_0a/\pi+O(1)$, and Conjecture~\ref{conj:law}: this needs the eigenvalues of $\psi_\kappa(-\Delta)$ on a growing interval at a fixed level with error $o(1/a)$, for a symbol of logarithmic growth, beyond what the half-line analysis of \cite{Kw1,KKM} gives. (2) Prove \eqref{eq:smooth}, for instance by extending Widom's theorem to symbols of logarithmic growth and proving $\chi_F=\mathcal B(\sigma;F)$, and the complement laws of Section~6, and find a dual count $h^1$ that vanishes for large degree, with a canonical $K$, that would make $h^0_F-h^1_F$ a Riemann--Roch formula in the proper sense. (3) Explain why the phase separates the parities at small levels and what the curves of Figure~\ref{fig:rr}(a) mean arithmetically. (4) Find an exact form of the conductor duality of Section~7, from the Gaussian rather than from Stirling, and see whether it combines with the complement duality of Section~6 into a single Riemann--Roch formula in which $\zeta$ is the self-dual point.

\section*{Data and code}
The scripts (NumPy, SciPy) are in the ancillary files: \texttt{rr\_common.py} (symbols, $t_0$, phase, Toeplitz entries, Levinson), \texttt{rr\_test\_fase.py} ($t_0$ and the predicted shifts), \texttt{rr\_scan.py} (all windows and jumps), \texttt{fase\_check.py} (second quadrature of the phase), \texttt{seno\_check.py} (sine basis), \texttt{traza\_suave.py}, \texttt{traza\_pocos.py}, \texttt{chi\_pred.py} (the smooth count; several minutes), \texttt{widom\_check.py} (Widom's second term, ten seconds), \texttt{circulo.py} (complements on a circle), \texttt{dualidad\_q.py} (conductor duality), \texttt{fig\_rr.py}, with the log \texttt{rr\_scan.jsonl}; the independent re-implementation is in the folder \texttt{verificacion\_independiente}. File names are in Spanish (\emph{fase} phase, \emph{traza} trace, \emph{seno} sine, \emph{suave} smooth).

\section*{Use of AI assistance}
The research programme is the author's. Claude (Anthropic) was used as a research and writing aid: to derive and check proofs, to write and run the computations, to search the literature, and to draft the text and figures. The main computation was re-implemented independently, from a separate derivation, as a check (Section~4). The author checked the results and is responsible for the content.

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\end{document}