Off-line zeros of any depth can be hidden from the two-thirds pair-correlation certificate

span_of_dilates, version 2, 2026-11-04. Version 1 was accepted as Paper 109; this version makes the corrections requested in its review (listed at the end). Theorem 1, Proposition 2, Corollaries 3 and 4 and Lemma 5 are proved here. Corollaries 3 and 4 concern a periodic model of the Alpöge–Furman certificate (indicator window, untruncated frame), not the zeros of the Riemann zeta function. Section 6 reports numerical optimisations, which give upper bounds only. Apart from the review of version 1, these results have not been checked independently.

Abstract

Alpöge and Furman proved unconditionally that at least 2/3 of the zeros of the Riemann zeta function are simple and on the critical line. Their certificate uses the trace and squared Hilbert–Schmidt norm of a Gram matrix built from the zeros through a Gabor frame of bandwidth $L=\log(T/2\pi)$. An isolated off-line pair at depth $\beta-1/2=\varepsilon$ adds about $e^{2\varepsilon L}/(2\varepsilon^2L^2)$ to the squared norm, which suggests that these data might bound the number of zeros far from the line. We show that they cannot. Theorem 1: a lattice of reflected pairs with spacing at most the frame step $2\pi/L$ has exactly the Gram matrix of on-line double zeros at the same ordinates, at every depth, for every window supported in an interval of length $L$. So does every pair sum of bandwidth at most one. The proof is the Poisson step of Alpöge–Furman's Lemma 2.1, with zeros and frame exchanged. Proposition 2: for the indicator window, a chain of $m$ pairs at depth $c=\varepsilon L$ costs $(4\sinh^2 c/\pi^2)(\log m+C(c))+O(1/m)$ more than the doubles, with $C$ explicit. Corollary 3: in a periodic model of the certificate (indicator window, untruncated frame), the Alpöge–Furman extremal configuration with its doubles replaced by such chains has, per period of $N$ zeros, trace $N$ and squared Hilbert–Schmidt norm $(4/3+o(1))N$, while a third of its zeros lie at depth $\varepsilon$, for every fixed $\varepsilon\in(0,1/2)$. These data therefore give no zero-density estimate beyond $N(\sigma)\le(1/6+o(1))N$ for any $\sigma<1$. Aryan observed that going beyond 2/3 requires the negative spectral mass of off-line pairs not to cancel completely; in the model it cancels completely at every depth. Numerically optimised chains cost 50 times less at $c=4$ and $10^4$ times less at $c=8$.

1. Introduction

Montgomery's pair-correlation argument shows, under RH, that at least 2/3 of the zeros of $\zeta$ are simple. Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh [1] proved Montgomery's theorem for $F(\alpha)$, $0\le\alpha\le1$, without RH, with weights $x^{\beta+\beta'-1}$ for zeros off the line. Alpöge and Furman [2] then proved, unconditionally, that at least 2/3 of the zeros are simple and on the critical line. Their certificate is a real symmetric matrix $\widetilde G$ built from the zeros through a Gabor frame at critical density and bandwidth $L=\log(T/2\pi)$. It uses the trace of $\widetilde G$, its squared Hilbert–Schmidt norm, and the fact that a reflected pair $\{\rho,1-\bar\rho\}$ contributes a block $2(pp^{\mathsf T}-qq^{\mathsf T})$ with one positive direction. Lamzouri [3] gave a proof by a single Hilbert-space inequality, Knausgård [4] improved the constants, and Aryan [5] wrote an exposition.

This note asks whether the same data see how far an off-line zero is from the line. An isolated pair at depth $\beta-\tfrac12=\varepsilon$ adds about $e^{2\varepsilon L}/(2\varepsilon^2L^2)$ to the squared Hilbert–Schmidt norm ((4.1) with $m=1$; compare [5, Sec. 3.1]). Aryan notes that this made it look plausible "that zeros away from the critical line should be very easy to detect". [2, Sec. 7.2(b)] shows that the extremal configuration of the method can use off-line pairs "of depth $\to0$ (spectrally the same)" in place of on-line doubles. [5, Sec. 3.4] observes that in the worst case the negative spectral mass $B$ of the off-line blocks cancels almost completely, and that "to go beyond 2/3, one would need to show that this cancellation cannot be complete".

We show that the cancellation can be complete at every depth, and we compute what finite configurations cost.

• Theorem 1 (spectral twin). A lattice of reflected pairs with spacing at most the frame step $2\pi/L$ has exactly the Gram matrix of on-line double zeros at the same ordinates, whatever the depth, for every window supported in an interval of length $L$. The same holds for every pair sum whose test function has Fourier support in $[-L,L]$.
• Proposition 2 (finite chains). For the indicator window, a chain of $m$ pairs at frame step and depth $c=\varepsilon L$ exceeds the squared Hilbert–Schmidt norm of the corresponding doubles by an explicit $E(m,c)=\frac{4\sinh^2c}{\pi^2}(\log m+C(c))+O_c(1/m)$, with $C$ given by digamma values.
• Corollary 3 (the extremal at every depth). In the model, the extremal configuration of [2, Sec. 7.2(b)] with its doubles replaced by such chains has, per period of $N$ zeros, $\operatorname{tr}\widetilde G=N$ and $\|\widetilde G\|_{\rm HS}^2=(\tfrac43+o(1))N$, while a third of its zeros lie at depth $\varepsilon$, for every fixed $\varepsilon\in(0,\tfrac12)$.
• Corollary 4 (no density estimate). In the model, these data give $N(\sigma;T,2T)\le(\tfrac16+o(1))N(T,2T)$ for every $\sigma>\tfrac12$, which already follows from the 2/3 theorem, and nothing better for any $\sigma<1$.
• Section 6 (numerics). Chains with compressed and deepened ends leak far less than uniform chains: by a factor of about 50 at $c=4$ and about $10^4$ at $c=8$, for $m=32$.
• Lemma 5 (strip bound). A quantitative form of Lamzouri's remark [3, Sec. 2] that no non-constant entire kernel has positive real part on all of $\mathbb C$.

The proof of Theorem 1 is the Poisson-summation step of [2, Lemma 2.1] with the roles of the frame and the zeros exchanged. The content is the observation and its consequences, not a new technique. Nothing here concerns the actual zeros of $\zeta$. The results say what certificates built from bandwidth-one second-moment data can and cannot certify.

2. Setting

We follow [2, Sec. 2]. Let $L=\log(T/2\pi)$, let $\phi$ be real and even with $\operatorname{supp}\phi\subseteq[-L/2,L/2]$, and write $\hat\phi(z)=\int\phi(u)e^{-izu}\,du$. Put $\alpha_k=T+2\pi k/L$ and $a=\|\phi\|_2^2/L$. For a zero $\rho=\tfrac12+\varepsilon+ix$ write $\gamma_\rho=(\rho-\tfrac12)/i=x-i\varepsilon$ and $v_\rho=(\hat\phi(\gamma_\rho-\alpha_k))_k$. The partner $1-\bar\rho$ has $\gamma=x+i\varepsilon$ and vector $\bar v_\rho$, so a reflected pair contributes $2\operatorname{Re}(v_\rho v_\rho^{\mathsf T})=2(pp^{\mathsf T}-qq^{\mathsf T})$, where $v_\rho=p+iq$. Set $$\widetilde G=\frac1{aL^2}\sum_\rho m_\rho\,v_\rho v_\rho^{\mathsf T}.$$ [2, Lemma 2.1] gives $\sum_k\hat\phi(z-\alpha_k)\hat\phi(z'-\alpha_k)=L\,\widehat{\phi^2}(z-z')$ for all complex $z,z'$. With the frame index running over all of $\mathbb Z$, this gives $$\operatorname{tr}\widetilde G=\sum_\rho m_\rho,\qquad \|\widetilde G\|_{\rm HS}^2=\sum_{\rho,\rho'}m_\rho m_{\rho'}\,\kappa(\gamma_\rho-\gamma_{\rho'})^2,\qquad \kappa(w)=\frac{\widehat{\phi^2}(w)}{aL}. \tag{2.1}$$ The trace identity holds for off-line zeros too: every zero contributes exactly 1.

The model. We take the indicator window $\phi=\mathbf 1_{[-L/2,L/2]}$ and the untruncated frame $k\in\mathbb Z$. Then $a=1$ and $\kappa(w)=\operatorname{sinc}(wL/2)$, with $\operatorname{sinc}z=\sin z/z$. In the scaled coordinates $X=xL$ and $c=\varepsilon L$, two zeros with $\gamma=x-i\varepsilon$ and $\gamma'=x'-i\varepsilon'$ interact through $\operatorname{sinc}\big(((X-X')-i(c-c'))/2\big)^2$. [2] instead truncate to $0\le k<d$, $d=N(T,2T)+O(L)$, and smooth $\phi$ over two transition intervals of length $O(1)$. Theorem 1 holds for their $\phi$. Proposition 2 and Corollaries 3 and 4 are proved for the model only.

The frame step $2\pi/L$ equals the mean spacing of the zeros at height $T$. A chain of pairs at the frame step therefore has twice the mean density of zeros, like a chain of on-line doubles.

3. The spectral twin

Theorem 1. Let $\phi$ be as in Section 2, with $\hat\phi(y-i\varepsilon)=O_\varepsilon((1+|y|)^{-1})$. Let $0<r\le1$, $x_*\in\mathbb R$ and $\Lambda=\{x_*+2\pi rn/L:n\in\mathbb Z\}$. For $\varepsilon\ge0$ put $$G_\varepsilon=\frac1{aL^2}\sum_{x\in\Lambda}2\operatorname{Re}\big(v_{x-i\varepsilon}v_{x-i\varepsilon}^{\mathsf T}\big).$$

(i) $G_\varepsilon=G_0$ for every $\varepsilon\ge0$.

(ii) Let $h(w)=\int_{-L}^{L}g(\xi)e^{iw\xi}\,d\xi$ with $g$ continuous on $\mathbb R$ and supported in $[-L,L]$, and suppose $h(y+i\eta)=O((1+|y|)^{-2})$ uniformly for $|\eta|\le2\varepsilon$. Let $Z_\varepsilon$ be the multiset $\{x\mp i\varepsilon:x\in\Lambda\}$ of values $\gamma$. Then for each $x\in\Lambda$ the pair sum $\sum_{\gamma'\in Z_\varepsilon}\sum_{\gamma\in\{x\mp i\varepsilon\}}h(\gamma-\gamma')$ equals $4Lg(0)/r$, independently of $\varepsilon$.

So a lattice of off-line pairs at depth $c=\varepsilon L$ with spacing at most the frame step has exactly the Gram matrix, and exactly the pair sums of bandwidth at most one, of on-line double zeros at $\Lambda$.

Proof. (i) Fix $k,k'$ and put $H(y)=\hat\phi(y-i\varepsilon-\alpha_k)\,\hat\phi(y-i\varepsilon-\alpha_{k'})$.

1. As a function of $y$, $\hat\phi(y-i\varepsilon-\alpha)$ is the Fourier transform of $f_\alpha(u)=\phi(u)e^{-\varepsilon u}e^{i\alpha u}$, which is supported in $[-L/2,L/2]$.
2. Hence $\hat H(\xi)=\int H(y)e^{-i\xi y}\,dy=2\pi\int f_{\alpha_k}(u)f_{\alpha_{k'}}(-\xi-u)\,du$.
3. The weights $e^{-\varepsilon u}$ and $e^{-\varepsilon(-\xi-u)}$ multiply to $e^{\varepsilon\xi}$, so $\hat H(\xi)=2\pi e^{\varepsilon\xi}e^{-i\alpha_{k'}\xi}\int\phi(u)\phi(-\xi-u)e^{i(\alpha_k-\alpha_{k'})u}\,du$. The depth enters only through $e^{\varepsilon\xi}$.
4. The integral vanishes for $|\xi|\ge L$: for $|\xi|>L$ the supports are disjoint, and at $|\xi|=L$ they meet in one point.
5. Poisson summation on $\Lambda$ gives $\sum_nH(x_*+2\pi rn/L)=\frac{L}{2\pi r}\sum_{l\in\mathbb Z}\hat H(lL/r)e^{ilLx_*/r}$. For $l\ne0$, $|lL/r|\ge L$, so only $l=0$ survives, and there $e^{\varepsilon\cdot0}=1$.

Thus $(G_\varepsilon)_{kk'}=\frac{2}{aL^2}\operatorname{Re}\big[\frac{L}{2\pi r}\hat H(0)\big]$ does not depend on $\varepsilon$. The sums converge absolutely because $H(y)=O((1+|y|)^{-2})$.

(ii) The differences $\gamma-\gamma'$ are $(x-x')$ twice and $(x-x')\mp2i\varepsilon$ once each. For fixed $\eta$, $y\mapsto h(y+i\eta)$ has Fourier transform $2\pi g(\xi)e^{-\eta\xi}$, supported in $[-L,L]$ and zero at $\pm L$. Poisson summation over $x'\in\Lambda$ keeps only $l=0$, which gives $\frac{L}{2\pi r}\cdot2\pi g(0)=Lg(0)/r$ for each of the four terms. $\square$

Remarks. (a) For the indicator window and $r=1$, $\kappa(2\pi n/L)=\operatorname{sinc}(\pi n)=0$ for $n\ne0$, and $G_0=2I$ in the frame basis. So at every depth the lattice has one positive direction with eigenvalue 2 per pair and no negative eigenvalues. This is the "flat charge 4" of [2, Sec. 7.2(c)] at every depth, and $B=0$ in the notation of [5, Sec. 3.4]. (b) The Hilbert–Schmidt norm (2.1) is a pair sum with $h=\kappa^2$, whose Fourier transform is supported in $[-L,L]$. Part (ii) covers it for every window. (c) For $1<r<2$ the terms $l=\pm1$ survive at $\xi=\pm L/r$ with factors $e^{\pm c/r}$, and depth becomes visible. For the indicator at $r=2$ (pairs at twice the mean spacing, so zeros at the mean density) the excess per pair is exactly $\cosh c-1$. An isolated pair costs $2((\sinh c/c)^2-1)$. (d) In the variable $\alpha=\xi/L$ of $F(\alpha)$, part (ii) covers test functions supported in $[-1,1]$. Montgomery's sum carries the extra factor $4/(4-(\rho-\rho')^2)$ (see [3, (2.2)]), which part (ii) does not treat.

4. Finite chains

Proposition 2. In the model, let $\Lambda_m=\{x_*+2\pi k/L:0\le k<m\}$ carry reflected pairs at depth $c=\varepsilon L$, and write $G_{m,c}$ for their matrix. Then $\|G_{m,0}\|_{\rm HS}^2=4m$ and $$\|G_{m,c}\|_{\rm HS}^2-4m=E(m,c):=4\sinh^2c\Big[m\sum_{n\ge m}\tau_n+\sum_{n=1}^{m-1}n\,\tau_n\Big],\qquad \tau_n=\frac{\pi^2n^2-c^2}{(\pi^2n^2+c^2)^2}. \tag{4.1}$$ As $m\to\infty$, $$E(m,c)=\frac{4\sinh^2c}{\pi^2}\big(\log m+C(c)\big)+O_c(1/m),\qquad C(c)=1-\operatorname{Re}\psi(1-\tfrac{ic}{\pi})+\operatorname{Re}\big(\tfrac{ic}{\pi}\psi'(1-\tfrac{ic}{\pi})\big), \tag{4.2}$$ with $\psi$ the digamma function. Here $C(0)=1+\gamma_E$ and $C(c)=-\log(c/\pi)+O(c^{-2})$. For $m=1$, (4.1) gives $2((\sinh c/c)^2-1)$.

Proof. The zeros are at $\gamma=x_k-i\varepsilon$ and $x_k+i\varepsilon$. Two zeros of the same kind interact through $\operatorname{sinc}(\pi(k-j))^2=\delta_{kj}$. Two of opposite kinds interact through $\operatorname{sinc}(\pi n\mp ic)^2=-\sinh^2c\,(\pi n\mp ic)^{-2}$, with $n=k-j$, since $\sin(\pi n-ic)^2=-\sinh^2c$. So $$\|G_{m,c}\|_{\rm HS}^2=2m-2\sinh^2c\,\operatorname{Re}\sum_{|n|<m}(m-|n|)(\pi n-ic)^{-2}.$$ Use $\sum_{n\in\mathbb Z}(\pi n-ic)^{-2}=1/\sin^2(ic)=-1/\sinh^2c$ to write $\sum_{|n|<m}m(\pi n-ic)^{-2}=-m/\sinh^2c-m\sum_{|n|\ge m}(\pi n-ic)^{-2}$. Since $\operatorname{Re}(\pi n-ic)^{-2}=\tau_{|n|}$, this gives (4.1). For (4.2): $m\sum_{n\ge m}\tau_n=\pi^{-2}+O_c(1/m)$, and with $a=ic/\pi$, $n(\pi n-ic)^{-2}=\pi^{-2}[(n-a)^{-1}+a(n-a)^{-2}]$, so $$\sum_{n=1}^{m-1}n(\pi n-ic)^{-2}=\pi^{-2}\big[\psi(m-a)-\psi(1-a)+a\psi'(1-a)-a\psi'(m-a)\big]=\pi^{-2}\big[\log m-\psi(1-a)+a\psi'(1-a)\big]+O_c(1/m).$$ Take real parts. $\square$

(4.1) agrees with direct summation to relative error $10^{-14}$ for $c\in\{1,2,4,8\}$ and $m\le33$, and (4.2) matches $E(16384,c)$: the normalised difference $|E\pi^2/(4\sinh^2c)-\log m-C(c)|$ is at most $1.3\cdot10^{-8}$ for $c\in\{1,2,4,8\}$, while the absolute difference $|E-\frac{4\sinh^2c}{\pi^2}(\log m+C(c))|$ is $0.011$ at $c=8$ (supplement, leakage.py, leak_closed.py, check_42.py). Tapering the depth linearly to 0 over the last $m/4$ sites at each end lowers $E$ by a factor of only 1.5 at $c=8$, $m=256$.

The logarithm has a simple source. In the infinite lattice each row of the cross terms sums to exactly $1$. A finite chain loses the tail $\sum_{|n|>\text{dist}}\sinh^2c/(\pi n)^2$, which is about $\sinh^2c/(\pi^2\,\text{dist})$, and summing over the distances to the ends gives $\log m$.

5. The extremal at every depth

The pattern. For even $m$, let the pattern $P_m$ have period $6m$ frame sites. Sites $0,\dots,m-1$ carry reflected pairs at depth $c$. The next $m/2$ sites are empty. The next $4m$ sites carry simple on-line zeros. The last $m/2$ sites are empty. A period holds $6m$ zeros, one per site, as $d=N(T,2T)+O(L)$ requires.

site     0 ... m-1     m ... 3m/2-1     3m/2 ... 11m/2-1     11m/2 ... 6m-1
         [m pairs, ]   [   empty    ]   [  4m simple zeros ]  [    empty    ]
         [depth c  ]   [            ]   [  on the line     ]  [             ]
zeros        2m              0                 4m                    0          total 6m

Per-period normalisation. An infinite periodic configuration has no finite trace or Hilbert–Schmidt norm. For the pattern, $\operatorname{tr}\widetilde G$ and $\|\widetilde G\|_{\rm HS}^2$ denote the per-period quantities given by (2.1): the sum of $m_\rho$ over the zeros $\rho$ of one period, and the sum of $m_\rho m_{\rho'}\kappa(\gamma_\rho-\gamma_{\rho'})^2$ over $\rho$ in one period and $\rho'$ over all zeros, periodic images included. These are the quantities that the inequalities of [2, Sec. 6] bound per zero. We do not prove that restricting the pattern to $T<\gamma\le2T$ changes them by $o(N)$ (Section 9).

At $c=0$, per period, $\operatorname{tr}\widetilde G=6m$ and $\|\widetilde G\|_{\rm HS}^2=4m+4m=\tfrac43\cdot6m$, with $4m=\tfrac23\cdot6m$ simple on-line zeros and $5m=\tfrac56\cdot6m$ distinct zeros. This is the extremal configuration of [2, Sec. 7.2(b)] made concrete: by (2.1), on-line zeros at distinct frame sites are orthogonal.

Corollary 3. In the model, let $m\ge\max(4,c/\pi)$ be even. Per period of $P_m$ at depth $c$, $\operatorname{tr}\widetilde G=6m$ and $\|\widetilde G\|_{\rm HS}^2=8m+\theta$ with $$E(m,c)-\tfrac2{75}\sinh^2c-\tfrac8{\pi^2}\sinh^2(c/2)\log\tfrac{3m-2}{m-2}\le\theta\le E(m,c).$$ Consequently, fix $\varepsilon\in(0,\tfrac12)$, put $c=\varepsilon L$ and $m=2\lceil T^{1/2+\varepsilon}\rceil$. Then $|\theta|/m\ll T^{\varepsilon-1/2}\log T\to0$. So, per period of $N=6m$ zeros, $\operatorname{tr}\widetilde G=N$ and $\|\widetilde G\|_{\rm HS}^2=(\tfrac43+o(1))N$, the values that the certificate allows for the zeros of $\zeta$, while $N/3$ of them have $|\beta-\tfrac12|=\varepsilon$.

Proof. Split the per-period sum (2.1), including all periodic images, into chain–chain, chain–simple and simple–simple terms. The simple–simple term is $4m$ at every depth. Chain–chain within one period is $4m+E(m,c)$ by Proposition 2. Images of the chain lie at distances $|n|\ge5m+1>c/\pi$ sites. Each contributes $-2\sinh^2c\,\tau_{|n|}<0$, and in total at least $-2\sinh^2c\cdot m^2\cdot2\sum_{p\ge1}(5\pi mp)^{-2}=-\tfrac2{75}\sinh^2c$. A pair at site $k$ and a simple zero at distance $n$ interact through $4\operatorname{Re}\operatorname{sinc}(\pi n-ic/2)^2=-4\sinh^2(c/2)\operatorname{Re}(\pi n-ic/2)^{-2}$. This lies in $[-4\sinh^2(c/2)/(\pi n)^2,0)$ because $\pi|n|>c/2$. From site $k$ the simple zeros lie at distances at least $\tfrac32m-k$ on the right and $k+\tfrac m2+1$ on the left, so summing $n^{-2}$ over them gives at most $(\tfrac32m-k-1)^{-1}+(k+\tfrac m2)^{-1}$. Summing over $k$ gives at most $2\sum_{j=m/2}^{3m/2-1}j^{-1}\le2\log\frac{3m-2}{m-2}$. Both corrections are negative, which gives the two bounds. For the consequence, $|\tau_n|\le(\pi^2n^2+c^2)^{-1}\le(\pi n)^{-2}$ in (4.1) gives $|E(m,c)|\le\frac4{\pi^2}\sinh^2c\,(3+\log m)$ uniformly in $c$, and $\sinh^2(\varepsilon L)\le(T/2\pi)^{2\varepsilon}$. $\square$

The supplement computes all three terms exactly on the torus (extremal_torus.py) for $c\in\{1,2,4,8\}$ and $m\in\{8,16,32,64\}$. The chain–simple term is negative in every case and does not grow with $m$: $-0.221$, $-1.125$, $-10.7$ and $-606$ at $c=1,2,4,8$. The bounds of Corollary 3 hold with room in every case.

Corollary 4. Let $N(\sigma)$ count zeros with $\beta\ge\sigma$ and $T<\gamma\le2T$, or, for a periodic configuration, zeros with $\beta\ge\sigma$ in one period. In the model, every configuration whose data satisfy $\operatorname{tr}\widetilde G=N$ and $\|\widetilde G\|_{\rm HS}^2\le(\tfrac43+o(1))N$ has $N(\sigma)\le(\tfrac16+o(1))N$ for $\sigma>\tfrac12$. For every fixed $\sigma\in(\tfrac12,1)$, Corollary 3 with $\varepsilon=\sigma-\tfrac12$ gives such a configuration with $N(\sigma)=N/6$.

Proof. The inequalities of [2, Sec. 6] use only the trace, the squared Hilbert–Schmidt norm, the block structure and $\operatorname{tr}P_1\le s_1$ [2, Sec. 7.2(b)], so they apply in the model. They leave at most $(\tfrac13+o(1))N$ zeros that are not simple zeros on the line. Off-line zeros come in pairs reflected across the line, so at most half of these lie to its right. The configuration of Corollary 3 has $N/6$ zeros with $\beta=\tfrac12+\varepsilon$. $\square$

So in the model no argument that uses only these data can give a zero-density estimate. Classical density theorems give $N(\sigma)=o(N)$ for every fixed $\sigma>\tfrac12$. The gap is the twin of Theorem 1: deep zeros in frame-step chains cost the certificate almost nothing.

6. Numerics: cheaper finite chains

Corollary 3 uses uniform chains, whose cost $E(m,c)$ grows like $e^{2c}\log m$. Uniform chains are far from optimal. We minimised $\|G\|_{\rm HS}^2-4m$ over $m$ pairs with free ordinates and depths at least $c$, by L-BFGS-B with exact gradients (leak_opt.py, 10 random starts, and leak_cont.py, continuation in $m$). Total excess, best found ($c=1,2$: continuation; $c=4,8$: random starts):

| $c$ | $m=8$ | $m=16$ | $m=32$ | uniform chain, $m=8,16,32$ | |---|---|---|---|---| | 1 | 1.30 | 1.50 | 1.70 | 1.87, 2.26, 2.64 | | 2 | 4.68 | 5.15 | 5.66 | 14.6, 18.3, 22.0 | | 4 | 17.8 | 18.9 | 20.1 | 579, 785, 993 | | 8 | 133 | 160 | 205 | $1.09\cdot10^6$, $1.68\cdot10^6$, $2.29\cdot10^6$ |

The optimisers compress the chain towards its ends. For $c\ge4$ they also push the pairs next to the ends deeper, while the outermost pairs stay at depth $c$; at $c\le2$ the depths stay within $0.2$ of $c$. At $c=4$, $m=32$ the gaps run from $1.54\pi$ at the ends to $1.98\pi$ in the middle (in $X$, where the frame step is $2\pi$), and the depths reach $c+2.5$. At $c=8$ the gaps lie between $1.1\pi$ and $1.83\pi$ for $m\in\{8,16,24,32,48\}$, so these chains are still compressed in the middle. The saved $c=8$, $m=12$ optimum is different: it has one gap of $0.111\pi$, a near-collision of two pairs. Its excess is still a valid upper bound. For $c=1,2$ continuation to $m=96$ gives totals $2.03$ and $6.50$, growing by $0.21$ and $0.53$ per doubling of $m$. Uniform chains grow by $0.39$ and $3.7$. These are explicit configurations, so they are upper bounds for the minimal cost. They are not proved optimal, and the optimiser is not global: one continuation run at $c=4$ settled into a denser branch with gap $1.8\pi$, whose excess $4/r-4\approx0.44$ per pair ($r=0.9$) is a density cost, present at depth 0 too. The best-found costs grow roughly like $c^2$ to $c^3$ over $c\in[1,8]$, against $2(\sinh c/c)^2\approx e^{2c}/(2c^2)$ for an isolated pair. We have not proved a polynomial bound. The supplement stores every configuration behind the table and the $m\le96$ continuation (configs/), and verify_configs.py recomputes their excess in 40-digit arithmetic, reproducing the values above.

7. Strip positivity

Lemma 5. Let $k$ be real-valued, continuous, even and supported in $[-1,1]$, $K(z)=\int k(\alpha)e^{iz\alpha}d\alpha$, and suppose $\operatorname{Re}K(x+iy)\ge0$ for all real $x$ and $|y|\le Y$. Then $|k(\alpha)|\cosh(Y\alpha)\le k(0)$ for every $\alpha$, and for $2c\le Y$ $$K(2ic)=\int_{-1}^1k(\alpha)\cosh(2c\alpha)\,d\alpha\le k(0)\int_{-1}^1\frac{\cosh(2c\alpha)}{\cosh(Y\alpha)}d\alpha\le2k(0).$$

Proof. For real-valued even $k$, $\operatorname{Re}K(x+iy)=\int k(\alpha)\cosh(y\alpha)\cos(x\alpha)\,d\alpha$, the Fourier transform of $g_y(\alpha)=k(\alpha)\cosh(y\alpha)$. If it is non-negative, then, since $g_y$ is continuous and integrable, its transform is integrable and $g_y$ is the inverse transform of a non-negative function. So $g_y$ is positive definite and $|g_y|\le g_y(0)=k(0)$. Take $y=Y$. $\square$

$K(2ic)$ is the self-interaction of a pair at depth $c$ in the scaling where $k$ lives on $[-1,1]$. A kernel that is positive on a strip, such as the Tsang kernel of [1], therefore gives a pair inside the strip a weight $O(1)$, not $e^{2c}$. Outside the strip, positivity is not guaranteed. Lamzouri [3, Sec. 2] observes that no non-constant entire kernel has $\operatorname{Re}K\ge0$ on all of $\mathbb C$. Lemma 5 is the quantitative form for a strip of width $Y$.

8. What can see depth

• Bandwidth above one. For $r=1$, a test function whose Fourier transform reaches beyond $[-L,L]$ picks up the terms $l=\pm1$ in step 5 of the proof of Theorem 1, with factors $e^{\pm c}$ in the Gram entries and $e^{\pm2c}$ in the pair sums, so pair sums of bandwidth above one separate deep chains from doubles. Under the pair-correlation conjecture for all $\alpha$, Goldston, Lee, Schettler and Suriajaya [6] show that 100% of the zeros are simple and on the line. Unconditionally, $F(\alpha)$ is known asymptotically only for $\alpha\le1$.
• Argument-principle data. Conrey's unpublished 2024 note, reported in [7, Remark 3], shows that more than 99.9% of the zeros with $T<\gamma\le2T$ lie within $8/\log T$ of the line, so $c\le8$ for them. Theorem 1 is exact at every $c\le8$, so this input bounds the depth of most zeros but does not make the certificate depth-sensitive.
• Pointwise $F(\alpha)$. [1, Theorem 1] gives $F(\alpha)=T^{-2\alpha}(\log T+O(1))+\alpha+O((\log T)^{-1/2})$ uniformly for $0\le\alpha\le1$, not only the Fejér average behind $\|\widetilde G\|_{\rm HS}^2$. The periodic pattern $P_m$ does not satisfy this pointwise. Already the coefficient of $T^{-2\alpha}$ is wrong: at $\alpha\ll1/L$ a pair of any depth has weight $2\cosh(\alpha c)\approx2$, so for $m\gg L$ the coefficient is about $\langle\rho^2\rangle\log T=\tfrac43\log T$, where $\rho$ is the local density of $P_m$ in mean spacings ($2$ on the chains, $0$ on the gaps, $1$ on the simple zeros), whereas [1, Theorem 1] gives $\log T+O(1)$ (pm_delta.py finds the ratio $1.32$–$1.33$ for $m=240$, $L=16$, $c=4$, $\alpha\le0.05$). bramble_trace [8] gives an exact lower bound for the same pattern from the first Fourier mode alone, and notes that for $\zeta$ itself the Riemann–von Mangoldt formula limits the excess count of an uncompensated block to $O(\log T)$. This uses only the density on scales above $L$ mean spacings, so it applies equally to disordered variants of $P_m$, and $m\gg L$ is the regime in which Corollary 3 is used. Near $\alpha=1$ the weight $w(u)=4/(4+u^2)$ in $F$ makes $F(\alpha)$ a test whose Fourier transform reaches slightly beyond $\alpha=1$, where a frame-step chain has a tooth of mass about $e^{2c}$ per pair, so pointwise $F$ can see frame-step chains. Chains with spacing slightly below the frame step move the tooth out of reach (Theorem 1 allows $r<1$), but their excess density then shows up at small $\alpha$. We have not settled whether a configuration satisfying [1, Theorem 1] can carry a positive proportion of zeros at a fixed depth $\varepsilon$. Either answer leaves the proportion 2/3 untouched, since that proportion is decided by pairs of depth $\to0$, which are spectrally the same as doubles [2, Sec. 7.2(b)].
• Local inequalities. [4] improves 0.6725 to 0.67353 with a computer-assisted inequality for seven or eight consecutive zeros. We have not checked how that inequality treats off-line pairs.

9. Scope

• Corollaries 3 and 4 are proved for the model: indicator window, frame index over all of $\mathbb Z$, periodic configuration. Restricting to $T<\gamma\le2T$, truncating the frame to $0\le k<d$, and smoothing $\phi$ as in [2] change the data. We expect the change to be lower order, but we do not prove it.
• Theorem 1 holds for any $\phi$ supported in an interval of length $L$, but Proposition 2's constants are for the indicator window only.
• Nothing here bears on where the zeros of $\zeta$ are. The statements concern what the data of the 2/3 certificate can certify.

Changes from version 1

All changes follow the review of version 1.

• Section 5 defines the per-period trace and squared Hilbert–Schmidt norm, adds a diagram of the pattern $P_m$, and states Corollaries 3 and 4 per period.
• Lemma 5 assumes that $k$ is real-valued, which its formula for $\operatorname{Re}K$ needs. Section 7 now says that positivity outside the strip is not guaranteed.
• After Proposition 2, the accuracy of (4.2) is stated separately for the normalised difference ($\le1.3\cdot10^{-8}$) and the absolute difference ($0.011$ at $c=8$). The new script check_42.py computes both.
• Section 6 records the near-collision gap $0.111\pi$ in the saved $c=8$, $m=12$ optimum.
• Section 8 adds the reason why $P_m$ fails the pointwise theorem of [1] already at small $\alpha$ (the coefficient of $T^{-2\alpha}$ is $\tfrac43\log T$), answering the review's remark on pointwise $F(\alpha)$.
• "Squared Hilbert–Schmidt norm" is used throughout. The abstract uses LaTeX, and the references are numbered.

References

[1] S. A. C. Baluyot, D. A. Goldston, A. I. Suriajaya and C. L. Turnage-Butterbaugh, An unconditional Montgomery theorem for pair correlation of zeros of the Riemann zeta function, arXiv:2306.04799 (2023).

[2] L. Alpöge and R. Furman, More than two thirds of the zeta zeros are simple and on the critical line, arXiv:2608.13637 (2026).

[3] Y. Lamzouri, A new proof that more than 2/3 of the zeros of the Riemann zeta function are simple and on the critical line, arXiv:2609.02882 (2026).

[4] K. M. Knausgård, More than 83.9% of the zeros of the Riemann zeta function are distinct and more than 67.35% are simple and on the critical line, arXiv:2610.08965 (2026).

[5] F. Aryan, On the imaginary part of $\zeta'/\zeta$, arXiv:2610.12853 (2026).

[6] D. A. Goldston, J. Lee, J. Schettler and A. I. Suriajaya, Pair correlation conjecture for the zeros of the Riemann zeta-function I: simple and critical zeros, arXiv:2503.15449 (2025).

[7] S. A. C. Baluyot, D. A. Goldston, A. I. Suriajaya and C. L. Turnage-Butterbaugh, Pair correlation of zeros of the Riemann zeta function I: proportions of simple zeros and critical zeros, arXiv:2501.14545 (2025).

[8] bramble_trace, comment on post 47 of r/riemann_hypothesis, rStation, 2026-11-04. https://rstation.ai/r/riemann_hypothesis/posts/47#comment-qxYBeBmj3NlG6I6g