Quantitative theta trial spaces for the Weil form

1. Contribution and comparison with prior work

We study the geometric Weil quadratic form on functions supported in $[-L,L]$. Weil positivity on every such window is equivalent to the Riemann hypothesis. Our results are unconditional estimates for explicitly constructed trial functions. They do not prove positivity of the form.

The fixed-window variational problem is classical: Bombieri [1] proves attainment of its minimum and analyzes the inertia of finite zero matrices. Connes and Consani [2, Section 3] explain small Weil energies through the global radical and construct spaces of approximate null vectors from prolate functions. Connes [3, Sections 6.3–6.6] describes double-exponential prolate decay and distinguishes those approximants from actual ground eigenvectors. These mechanisms and the broad decay scale are prior work.

We give a direct quantitative construction from the inverse Fourier transform of $\Xi$. For one specified shrinking cubic cutoff we prove:

1. An explicit absolute Rayleigh bound $10^{16}\exp(11L-2\pi e^{2L})$ for every $L\ge1$.
2. An actual trial-energy asymptotic $cL\exp(7L-2\pi e^{2L})$, with an explicit positive constant $c$.
3. Upper bounds at the same double-exponential scale for every fixed-index eigenvalue, using finite spans of theta derivatives.

The second result separates the archimedean leading term from the complete prime and pole interactions. Those interactions are estimated directly; no prime-number theorem, zero data, or RH assumption is needed.

For comparison, Zhu [4, Theorem 1.3] proves under RH the eventual upper bound $\lambda_*(L)\le\exp(-Le^L)$ for the even window infimum. Our construction supplies an unconditional upper bound at the $e^{2L}$ scale. It does not improve the much smaller certified finite-window values in [4], identify the asymptotic minimum, or establish that paper's fitted coefficient $2\pi^2$. Nor does it assert that the theta trial approximates a ground eigenvector.

2. Form, kernel, and statements

Use $\widehat f(z)=\int_{\mathbb R}f(x)e^{-izx}\,dx$ and $C_f(u)=\int_{\mathbb R}f(x+u)\overline{f(x)}\,dx$. Set

$$ \begin{aligned} Q(f)&=\frac1{2\pi}\int_{\mathbb R}h(t)|\widehat f(t)|^2\,dt +2\Re\!\left(\widehat f(i/2)\overline{\widehat f(-i/2)}\right)\\ &\hspace{12mm}-2\sum_{n\ge2}\frac{\Lambda(n)}{\sqrt n} \Re C_f(\log n),\\ h(t)&=\Re\psi(1/4+it/2)-\log\pi . \end{aligned} \tag{2.1} $$

Here $\Lambda$ is the von Mangoldt function and $\psi=\Gamma'/\Gamma$. For compact support the prime sum is finite. This is the normalization of [4, equations (2)–(3)], with the general complex pole pairing retained.

Let $P$ be the positive even theta kernel, given for $x\ge0$ by

$$ P(x)=\sum_{n\ge1} \left(4\pi^2n^4e^{9x/2}-6\pi n^2e^{5x/2}\right) e^{-\pi n^2e^{2x}}. \tag{2.2} $$

Its Fourier transform is $\Xi(z)=\xi(1/2+iz)$. This standard theta representation follows by applying $\partial_x^2-1/4$ to $e^{x/2}\sum_{n\ge1}e^{-\pi n^2e^{2x}}$, using theta inversion to obtain evenness, and integrating twice by parts in the theta integral for $\xi$. All derivatives of $P$ decay superexponentially at both ends. Positivity follows term by term in (2.2), since $\pi n^2e^{2x}>3/2$.

For $L\ge1$ put

$$ \delta=e^{-2L},\qquad a=L-\delta. $$

Define the even cutoff $\chi_L$ to be one on $[-a,a]$, zero outside $[-L,L]$, and

$$ \chi_L(x)=1-3s^2+2s^3,\qquad s=\frac{|x|-a}{\delta} $$

on $a<|x|<L$. Thus $\chi_L$ is $C^1$, piecewise smooth, and $|\chi_L'|\le3/(2\delta)$. Write $f_L=\chi_LP$.

Theorem 2.1 (explicit bound). For every $L\ge1$,

$$ \frac{|Q(f_L)|}{\|f_L\|_2^2} \le 10^{16}\exp(11L-2\pi e^{2L}). \tag{2.3} $$

In particular, the even window infimum obeys the same upper bound.

Define

$$ w(y)= \begin{cases} 0,&y\le-1,\\ 3(y+1)^2-2(y+1)^3,&-1<y<0,\\ 1,&y\ge0, \end{cases} \qquad I_0=\int_{-1}^{\infty}w(y)^2e^{-4\pi y}\,dy. $$

Theorem 2.2 (energy of the specified trial). As $L\to\infty$,

$$ \frac{Q(f_L)}{\|f_L\|_2^2} =\frac{64\pi^4I_0}{\|P\|_2^2}\, L e^{7L-2\pi e^{2L}}\left(1+O(L^{-1})\right). \tag{2.4} $$

The constant is positive, so this trial has positive energy eventually. The assertion is about its Rayleigh value, not the minimum.

Let $\lambda_m(L)$ denote the eigenvalues in increasing order, with multiplicity, of the closed form on the full complex window space. Write $\lambda_m^{\mathrm{ev}}$ and $\lambda_m^{\mathrm{odd}}$ for its parity restrictions.

Theorem 2.3 (fixed-dimensional trial spaces). For each fixed $m\ge1$, there are finite constants $C_m,L_m$ such that, for $L\ge L_m$,

$$ \lambda_m(L)\le C_m e^{(4m+7)L-2\pi e^{2L}}. \tag{2.5} $$

The parity restrictions satisfy corresponding bounds with linear exponents $8m+3$ and $8m+7$, respectively. More precisely, these bounds hold for the absolute Rayleigh form on explicit $m$-dimensional trial spaces. No uniformity for $m$ growing with $L$ is asserted.

For completeness, the window operator in this statement exists unconditionally. The multiplier $h$ is bounded below and tends to $+\infty$. Its closed form on zero-extended window functions has domain given by the finite logarithmically weighted Fourier norm. The finite prime shifts and the two pole functionals are bounded perturbations at fixed $L$. Fourier truncation shows that its form-domain embedding into the physical $L^2$ space is compact: high-frequency mass tends uniformly to zero on a bounded form ball, and the Fourier transform between two bounded intervals is compact. Hence the operator has compact resolvent and the min–max principle applies. This is the same variational setting as [1, Section 4] and [2, Section 2].

3. Continuity and unconditional annihilation

Put

$$ X=H^1(\mathbb R)\cap L^2(\mathbb R,e^{2|x|}\,dx),\qquad N(g)^2=\|g'\|_2^2+\|e^{|x|}g\|_2^2. $$

Lemma 3.1. The Hermitian form $Q$ extends continuously to $X$, and

$$ |Q(g)|\le20N(g)^2. \tag{3.1} $$

Proof. The convergent digamma series gives

$$ h(t)=h(0)+\sum_{n\ge0} \frac{(t/2)^2}{(n+1/4)((n+1/4)^2+(t/2)^2)}. $$

Here $h(0)=-\gamma-\pi/2-3\log2-\log\pi$ lies between $-6$ and zero, and

$$ \sum_{n\ge0}(n+1/4)^{-3} <64+\frac{64}{125}+\frac8{25}<65. $$

Thus $|h(t)|\le6+(65/4)t^2$. The archimedean term has absolute value at most $6\|g\|_2^2+(65/4)\|g'\|_2^2$.

Each pole functional has absolute value at most $\sqrt2\|e^{|x|}g\|_2$, by Cauchy–Schwarz. The pole contribution is therefore at most four times that weighted squared norm.

The triangle inequality $|x|+|x+u|\ge|u|$ gives

$$ |C_{f,g}(u)|\le e^{-|u|} \|e^{|x|}f\|_2\|e^{|x|}g\|_2. \tag{3.2} $$

Also, since $\log x\,x^{-3/2}$ decreases for $x\ge2$,

$$ \sum_{n\ge2}\frac{\log n}{n^{3/2}} \le \frac{4+(5/2)\log2}{\sqrt2}<5. $$

Using $\Lambda(n)\le\log n$, the absolute prime contribution is at most ten times the weighted squared norm. As that norm dominates $\|g\|_2$, summing the estimates proves (3.1). Polarization gives continuity. Smooth compact functions are dense in $X$ by truncation and mollification. $\square$

Lemma 3.2. For every nonnegative integer $j$ and every $g\in X$,

$$ Q(P^{(j)},g)=0. \tag{3.3} $$

Proof. Start with smooth compact $g$. The polarized explicit formula [1, Sections 2–3] applied to the correlation with $P^{(j)}$ has a spectral factor $(iz)^j\Xi(z)$. It vanishes at every complex zeta ordinate $z=(\rho-1/2)/i$, with multiplicities included.

To justify the noncompact theta test, approximate it by smooth time cutoffs. The geometric terms converge by the superexponential tails and Lemma 3.1. Repeated integration by parts shows that the transforms of the omitted tails tend to zero with any prescribed inverse-power decay, uniformly on $|\Im z|\le1/2$. The usual $O(T\log(T+2))$ zero count then justifies passage through the spectral sum. This proves (3.3) for compact smooth $g$; density and continuity extend it to $X$. No positive sampling formula has been used. $\square$

In particular, with $r_L=(1-\chi_L)P$,

$$ Q(f_L)=Q(P-r_L)=Q(r_L). \tag{3.4} $$

The functions in (3.4) belong to $X$. The compact trial belongs to $H^1_0(-L,L)$, and even smooth interior approximations converge to it in $X$. Thus it supplies an upper bound even if the infimum is initially defined using only smooth tests.

4. Explicit estimate

For $x\ge0$, put $R=\pi e^{2x}$. Complete-theta estimates give

$$ 0<P(x)\le8\pi^2e^{9x/2-R},\qquad |P'(x)|\le106\pi^3e^{13x/2-R}. \tag{4.1} $$

Indeed, after factoring the first mode, both $\sum_{n\ge1}n^4e^{-(n^2-1)R}$ and $\sum_{n\ge1}n^6e^{-(n^2-1)R}$ are less than two for $R\ge\pi$. For exponent $k=4,6$, the tail begins at $2^ke^{-3\pi}$ and its successive ratio is bounded by $(3/2)^ke^{-5\pi}<1$. The derivative polynomial is $-8R^3+30R^2-15R$, whose modulus is at most $53R^3$ for $R\ge\pi$. This proves (4.1) with all theta modes included.

Let $A=\pi e^{2a}$. For $L\ge1$, $A>17$, and for $0\le\nu\le11/2$,

$$ \int_A^\infty R^\nu e^{-2R}\,dR\le A^\nu e^{-2A}. $$

The inequality follows from the logarithmic derivative: $2-\nu/R>1$. Substitution in (4.1) gives

$$ \int_a^\infty P'(x)^2\,dx\le5618\pi^5e^{11a-2A},\qquad \int_a^\infty P(x)^2\,dx\le32\pi^3e^{7a-2A}. $$

Using both half-lines and $|r_L'|^2\le2P'^2+2|\chi_L'|^2P^2$,

$$ \|r_L'\|_2^2\le D e^{11L-2A},\qquad D=22472\pi^5+288\pi^3. \tag{4.2} $$

The factors before the two integrals are $4$ and $9\delta^{-2}$. Similarly,

$$ \|e^{|x|}r_L\|_2^2\le64\pi^3e^{9a-2A}. \tag{4.3} $$

Since $e^{2L}\delta=1$ and $1-e^{-2\delta}\le2\delta$,

$$ -2A\le-2\pi e^{2L}+4\pi. \tag{4.4} $$

Finally $\chi_L=1$ on $[-1/10,1/10]$, and the first positive theta term gives

$$ \|f_L\|_2^2 \ge\frac15\left[4\pi(\pi-3/2)e^{-\pi e^{1/5}}\right]^2 >\frac1{100}. \tag{4.5} $$

Equations (3.1), (3.4), and (4.2)–(4.5) prove Theorem 2.1, since

$$ 100\cdot20(D+64\pi^3)e^{4\pi}<10^{16}. \tag{4.6} $$

The finite inequalities here, including $A(1)>17$, (4.5), and (4.6), are verified by outward Arb arithmetic in the supplement. The norm lower bound in (4.5) exceeds $0.03954$.

5. Energy asymptotic of the cubic cutoff

Put $p_L=P(L)$ and

$$ u_L(y)=w(y)\frac{P(L+\delta y)}{p_L}\quad(y\ge-1), \qquad u_L(y)=0\quad(y<-1). $$

The two tail supports are disjoint, and

$$ r_L(x)=p_L\left[ u_L((x-L)/\delta)+u_L((-x-L)/\delta)\right]. \tag{5.1} $$

5.1. Profile estimates

The theta series implies

$$ \begin{aligned} p_L&=4\pi^2e^{9L/2-\pi/\delta}(1+O(\delta)),\\ \|u_L\|_2^2&=I_0+O(\delta),\qquad \sup_{L\ge1}\|u_L\|_{W^{1,1}}<\infty. \tag{5.2} \end{aligned} $$

Here are details for the needed uniform estimates. The first-mode ratio is

$$ e^{(9/2)\delta y-(\pi/\delta)(e^{2\delta y}-1)} \frac{1-\frac{3\delta}{2\pi}e^{-2\delta y}} {1-\frac{3\delta}{2\pi}}. $$

It tends to $e^{-2\pi y}$. On $y\ge0$, use $e^{2\delta y}\ge1+2\delta y$ and

$$ 0\le e^{2\delta y}-1-2\delta y \le2\delta^2y^2e^{2\delta y}. $$

The resulting differences are bounded by $\delta$ times an integrable polynomial times a fixed decreasing exponential. The interval $[-1,0]$ is compact. The remaining theta modes begin with the relative factor $e^{-3\pi e^{2a}}$ and obey the same complete-series bounds as (4.1). This proves the $L^2$ estimate in (5.2).

A uniform envelope convenient below is

$$ P(a+\delta z)\le C p_L e^{-3z},\qquad z\ge0,\ L\ge1. \tag{5.3} $$

Indeed, $\delta<1/7$, $P(L)\ge3\pi^2e^{9L/2-\pi/\delta}$, and convexity of the exponential makes the slope in $z$ at most $(9/2)\delta-2\pi e^{-2\delta}<-3$. The derivative bound in (4.1), after multiplication by $\delta$, has the same property with $13/2$ in place of $9/2$. Together with the bounded, compactly supported derivative of $w$, this proves the uniform $W^{1,1}$ assertion in (5.2).

In particular,

$$ \|r_L\|_2^2=2p_L^2\delta(I_0+O(\delta)). \tag{5.4} $$

5.2. Archimedean contribution

Let $U_L=\widehat u_L$. Uniform $W^{1,1}$ bounds give $|U_L(s)|\le C/(1+|s|)$, while (5.1) gives

$$ \widehat r_L(t)=p_L\delta \left[e^{-itL}U_L(\delta t)+e^{itL}U_L(-\delta t)\right]. $$

The standard digamma asymptotic is $h(t)=\log(|t|/(2\pi))+O(|t|^{-1})$ for $|t|\ge1$, and $h$ is bounded on $[-1,1]$. Consequently,

$$ \int_{\mathbb R} \frac{|h(s/\delta)-\log(1/\delta)|}{(1+|s|)^2}\,ds=O(1). \tag{5.5} $$

For $|s|\le\delta$ the integral is $O(\delta\log(1/\delta))$; elsewhere use $O(1+|\log|s||)$ with the displayed weight. Subtract the constant $\log(1/\delta)$ in the Fourier integral and apply (5.5). Plancherel handles that constant exactly, including the two-packet cross term. We obtain

$$ Q_{\mathrm{arch}}(r_L) =2L\|r_L\|_2^2+O(p_L^2\delta). \tag{5.6} $$

5.3. Complete prime and pole contributions

For $u\ge0$, (5.3) and the two tail supports imply

$$ 0\le C_{r_L}(u)\le C^2p_L^2\left[ \frac{\delta}{3}e^{-3u/\delta} +(u-2a)e^{-3(u-2a)/\delta}\mathbf1_{\{u\ge2a\}} \right]. \tag{5.7} $$

The first term sums the two same-tail overlaps. For the opposite-tail overlap, the integration interval is $[a-u,-a]$ when $u\ge2a$. The two exponential slopes cancel there, leaving the second term.

Using $\Lambda(n)\le\log n$, the same-tail prime contribution is $O(p_L^2\delta e^{-3\log2/\delta})$. For the other contribution set $n_0=e^{2a}$ and $\alpha=3/\delta+1/2$. For $k=1,2$, the function

$$ F_k(x)=\log(x/n_0)^k(x/n_0)^{-\alpha},\qquad x\ge n_0, $$

is unimodal. Its sum over integer points is bounded by its integral plus twice its supremum. These equal $n_0 k!/(\alpha-1)^{k+1}$ and $(k/(e\alpha))^k$. It follows that

$$ \begin{aligned} &\sum_{n\ge n_0} \frac{\log n}{\sqrt n}\log(n/n_0)(n/n_0)^{-3/\delta}\\ &\quad\le e^a\left[\frac{2a}{(\alpha-1)^2} +\frac2{(\alpha-1)^3}\right] +2e^{-a}\left[\frac{2a}{e\alpha} +\frac4{e^2\alpha^2}\right]\\ &\quad=O((L+1)\delta e^{-L}). \end{aligned} \tag{5.8} $$

Thus the entire prime contribution has absolute value $O(p_L^2\delta(L+1)e^{-L})$. No estimate for primes in short intervals has been used.

Likewise, (5.3) bounds the pole functional by

$$ \int r_L(x)\cosh(x/2)\,dx \le\frac{2Cp_Le^{a/2}\delta}{3-\delta/2}. $$

Its squared pole contribution is $O(p_L^2\delta e^{-L})$. Combining these estimates with (5.4) and (5.6) gives

$$ Q(f_L)=Q(r_L)=4Lp_L^2\delta I_0+O(p_L^2\delta). \tag{5.9} $$

Finally, (5.3) gives $\|f_L\|_2^2=\|P\|_2^2+O(p_L^2\delta)$. Substitution of (5.2) into (5.9) proves Theorem 2.2.

6. Finite derivative spaces

For $x\ge0$ and $R=\pi e^{2x}$ write

$$ P^{(j)}(x)=e^{x/2}\sum_{n\ge1}R_j(n^2R)e^{-n^2R}, $$

where

$$ R_0(z)=4z^2-6z,\qquad R_{j+1}(z)=(1/2-2z)R_j(z)+2zR_j'(z). $$

The polynomial $R_j$ has degree $j+2$. Let $B_j$ be its absolute coefficient sum and set

$$ b_j=B_j\pi^{j+2} \sum_{n\ge1}n^{2j+4}e^{-(n^2-1)\pi}<\infty. $$

Then

$$ |P^{(j)}(x)|\le b_j e^{(2j+9/2)x-\pi e^{2x}}. \tag{6.1} $$

For fixed $j,k\ge0$, elementary exponential-tail integration gives, once $A=\pi e^{2a}$ is sufficiently large,

$$ \int_a^\infty e^{kx}|P^{(j)}(x)|^2\,dx \le\frac{b_j^2}{2\pi}e^{(4j+7+k)a-2A}. \tag{6.2} $$

Indeed the substituted power of $R$ is $2j+7/2+k/2$, and the argument used in Section 4 applies. For $r_j=(1-\chi_L)P^{(j)}$, (6.2) yields

$$ N(r_j)^2\le d_j e^{(4j+11)L-2A},\qquad d_j=\frac{2b_{j+1}^2+(11/2)b_j^2}{\pi}. \tag{6.3} $$

The derivative estimate uses $4\int_a^\infty|P^{(j+1)}|^2+ 9\delta^{-2}\int_a^\infty|P^{(j)}|^2$. The weighted norm contributes $2\int_a^\infty e^{2x}|P^{(j)}|^2$, whose linear exponent is smaller by two.

Let $J$ be any finite set of derivative orders, $|J|=m$, $q=\max J$. The functions $P^{(j)}$, $j\in J$, are linearly independent: Fourier transformation of a dependence gives a polynomial times $\Xi$ identically zero. Let $g_J>0$ be the smallest eigenvalue of their $L^2$ Gram matrix. Dominated convergence implies that, for all sufficiently large $L$,

$$ \left\|\chi_L\sum_{j\in J}c_jP^{(j)}\right\|_2^2 \ge\frac{g_J}{2}\sum_{j\in J}|c_j|^2. \tag{6.4} $$

Lemma 3.2 identifies the form of this compact vector with the form of $\sum c_jr_j$. Cauchy–Schwarz for the norm $N$, (3.1), (6.3), (6.4), and (4.4) therefore give

$$ \frac{\left|Q\!\left(\chi_L\sum c_jP^{(j)}\right)\right|} {\left\|\chi_L\sum c_jP^{(j)}\right\|_2^2} \le\frac{40e^{4\pi}}{g_J}\left(\sum_{j\in J}d_j\right) e^{(4q+11)L-2\pi e^{2L}}. \tag{6.5} $$

Apply min–max to the $m$-dimensional cutoff space. Taking $J=\{0,\ldots,m-1\}$ proves (2.5); taking the first $m$ even or odd derivative orders proves the parity statements. The sign-independent prime estimate in Lemma 3.1 is essential here: the derivative tails need not be nonnegative.

7. Scope and reproduction

The results construct small absolute form values on explicit spaces. For an indefinite operator that does not imply small operator residuals or two-sided bounds on its eigenvalues. Theorems 2.1 and 2.3 are spectral upper bounds only. Theorem 2.2 proves eventual positivity of its specified trial, not positivity of other tests.

The global radical, near-radical constructions, and double-exponential prolate scales in [2–3] are not new claims here. No convergence of selected ground vectors to the Riemann kernel, no new zero-free region, and no proof of RH is obtained. In particular, the result does not supply the lower comparison needed to exclude a negative direction outside the trial spaces. Bombieri's discussion of vanishing negative coordinate eigenvalues in spectral limits [1, Sections 10–13] remains relevant to that distinction.

The proofs above are analytic. The supplement contains two short CPU scripts, under the MIT license, requiring Python, python-flint and mpmath:

python revival126_theta_window_constants.py
python revival128_theta_tail_profile.py

They run from the supplement directory and create a results subdirectory. The first checks all finite constants in Section 4 by 160-bit outward Arb arithmetic. The second independently compares an exponential-polynomial formula for $I_0$ with Acb integration, and computes $\|P\|_2^2$ with explicit full-theta and spatial tails. The outputs give

$$ I_0=109.49104226458724886008\ldots,\qquad \frac{64\pi^4I_0}{\|P\|_2^2} =2134740.2587882873363178\ldots . $$

The numerical coefficient is not needed for the asymptotic proof. Three floating calculations of the rescaled tail norm check its normalization; they do not evaluate a Weil eigenvalue. No zero data, GPU computation, or large numerical search is used.

References

[1] E. Bombieri. Remarks on Weil's quadratic functional in the theory of prime numbers, I. Rendiconti Lincei, Matematica e Applicazioni 11 (2000), 183–233. http://www.bdim.eu/item?id=RLIN_2000_9_11_3_183_0

[2] A. Connes and C. Consani. Spectral triples and zeta-cycles. Enseignement Mathématique 69 (2023), 93–148; preprint 2021. https://arxiv.org/abs/2106.01715

[3] A. Connes. The Riemann Hypothesis: Past, Present and a Letter Through Time. 2026. https://arxiv.org/abs/2602.04022v1

[4] X. Zhu. Weil positivity in compact windows: a finite reduction, certified two-sided bounds, and a Landau–Widom decay law. Version 2, 2026. https://arxiv.org/abs/2608.24827v2