REVIEW
A Hardy-space operator acting on $\zeta$-zeros
Functional Analysis · Submitted 2 Oct 2026 · 7 references
Pith's verdict: A genuinely new bounded H2 operator whose adjoint vertically translates the Nyman-Beurling kernels, with a correct invariant-subspace reformulation of RH — the printed equivalence needs the N-bar/N conflation fixed, but the substance holds.
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Referee report
The paper
The paper claims
This paper constructs a bounded Hardy-space operator whose adjoint translates zeta kernels vertically, turning the Riemann hypothesis into the single containment $T_\tau h_k \in \mathcal{N}$; an off-line zero would force an infinite…
The reading
What carries the argument
The engine is the pair of identities linked to the Riesz kernels $k_s$: the vertical-translation formula $T^*_\tau k_s = \frac{\bar s + i\tau}{\bar s}\, k_{s - i\tau}$, which moves a zero's kernel down the strip by $-i\tau$, and the orthogonality criterion $k_s \in \mathcal{N}^\perp \iff \zeta(s) = 0$, which re-reads each landing point as a zeta zero. The operator itself is $T_\tau = D_\tau + C Q_\tau$, where $D_\tau e_j = (j+1)^{i\tau} e_j$ is unitary, $C$ is the Copson operator, and $Q_\tau$ is a diagonal with $\|Q_\tau\| \le |\tau|$, so $\|T_\tau\| \le 1 + 2|\tau|$. The relation $\Delta T_\tau = D_\tau \Delta$ with $\Delta = I - S^*$ makes $T_\tau$ quasi-conjugate to a diagonal operator and yields the one-parameter group law $T_\tau T_\sigma = T_{\tau + \sigma}$. Section 6 records the common eigenvectors $W^*_n k_s = n^{1 - \bar s} k_s$ for the operators $W_n$, which force any operator sharing the translation action to satisfy the intertwining relation $W_n T = n^{i\tau} T W_n$.
Extended reading notes
Core claim
The central discovery is the adjoint translation identity (Proposition 4.1): for $\tau \in \mathbb{R}$ and $\Re s > 1/2$, the operator $T_\tau = D_\tau + C Q_\tau$ satisfies $T^*_\tau k_s = \frac{\bar s + i\tau}{\bar s}\, k_{s - i\tau}$ and $\|T_\tau\| \le 1 + 2|\tau|$. Paired with the pairings $\langle h_k, k_s\rangle = -\frac{\zeta(s)}{s}(k^{1-s} - 1)$ and the kernel criterion $k_s \in \mathcal{N}^\perp \iff \zeta(s) = 0$, invariance of $\mathcal{N}$ under $T_\tau$ forces: if $\rho$ is any zero with $\Re \rho > 1/2$, then $\zeta(\rho - i n \tau) = 0$ for every $n \ge 0$, an arithmetic progression of zeros whose ordinates grow in steps of $|\tau|$, contradicting Ingham's bound $N(\sigma, T) = o(T)$. Theorem 5.1 asserts the resulting equivalence, RH holds iff $T_\tau h_k \in \mathcal{N}$ for some fixed $k \ge 2$, and supplies the quantitative gap $\mathrm{dist}(T_\tau h_k, \mathcal{N}) \ge |\zeta(\rho - i\tau)|\, |k^{1 - \rho + i\tau} - 1| / (|\rho|\, \|k_\rho\|) > 0$ whenever $\zeta(\rho - i\tau) \ne 0$ for an off-line zero $\rho$.
Load-bearing premise
The equivalence rests on identifying invariance of the algebraic span $\mathcal{N}$ with invariance of its closure $\overline{\mathcal{N}}$: the abstract invokes the closure, while the theorem's proof of (i)$\Rightarrow$(ii) writes '(A) gives $\mathcal{N} = H^2$' even though the density criterion (A) yields only $\overline{\mathcal{N}} = H^2$; if that identification fails, the three-way equivalence is not established as printed, although the single-vector direction (iii)$\Rightarrow$(i) still derives RH from $T_\tau h_k \in \mathcal{N}$.
Editorial extensions
If this is right
- If the single-vector condition $T_\tau h_k \in \mathcal{N}$ holds for some fixed $k \ge 2$ and $\tau \ne 0$, then the Riemann hypothesis follows; Theorem 5.1 reduces the whole question to one containment check.
- The quantitative bound (21) says that near a hypothetical off-line zero $\rho$ with $\zeta(\rho - i\tau) \ne 0$, the distance from $T_\tau h_k$ to $\mathcal{N}$ is bounded below by an explicit positive number, an obstruction any proof of membership must overcome.
- Any zero $\rho$ with $\Re \rho > 1/2$ would, under the invariance hypothesis, spawn zeros at $\rho - i n \tau$ for every $n$, so disproving RH by exhibiting one such zero would simultaneously exhibit an infinite arithmetic progression of zeros, the very structure Ingham's bound forbids.
- Because the family $(T_\tau)_{\tau \in \mathbb{R}}$ is a one-parameter group of invertible operators, the criterion is stable under the choice of $\tau$: verifying it for one nonzero $\tau$ settles it for all.
Reading between the lines
- The printed proof of direction (i)$\Rightarrow$(ii) reads the density criterion (A) as giving $\mathcal{N} = H^2$, but (A) yields only density of the span; the algebraic span of countably many vectors is a proper dense subspace. The three-way equivalence stands as printed only if (ii) is read as invariance of the closure $\overline{\mathcal{N}}$, exactly as the abstract frames it.
- The mechanism needs only the boundedness of $T_\tau$ plus the two quoted inputs, the density criterion and the kernel orthogonality criterion; the same construction should carry to other reproducing-kernel Hilbert spaces, such as weighted $H^2$ or half-plane variants, wherever an analogue of $\langle h_k, k_s\rangle = -\zeta(s)(k^{1-s} - 1)/s$ holds.
- The intertwining relation $W_n T = n^{i\tau} T W_n$ pins down the operator's form once the translation action on the $k_s$ is fixed; any alternative operator proposed for such an RH criterion must satisfy this relation, which is directly checkable against the canonical basis.
Original abstract
A Hardy space approach to the Nyman-Beurling and B\'aez-Duarte criterion for the Riemann Hypothesis (RH) was introduced in [7]. It states that the RH holds if and only if the span of a particular sequence of functions $(h_k)_{k\geq2}$, denoted $\mathcal{N}$, is dense in the Hardy space $H^2$. In this note we construct a bounded operator $T$ on $H^{2}$ such that if the closure of $\mathcal{N}$ is $T$-invariant, then either the RH holds or there exists an infinite progression of $\zeta$-zeros. In particular, this yields an operator-theoretic criterion for the Riemann hypothesis as an invariant subspace question.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
A Hardy space approach to the Nyman-Beurling and B\'aez-Duarte criterion for the Riemann Hypothesis (RH) was introduced in [7]. It states that the RH holds if and only if the span of a particular sequence of functions $(h_k)_{k\geq2}$, denoted $\mathcal{N}$, is dense in the Hardy space $H^2$. In this note we construct a bounded operator $T$ on $H^{2}$ such that if the closure of $\mathcal{N}$ is $T$-invariant, then either the RH holds or there exists an infinite progression of $\zeta$-zeros. In particular, this yields an operator-theoretic criterion for the Riemann hypothesis as an invariant subspace question.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A Hardy space approach to the Nyman-Beurling and B\'aez-Duarte criterion for the Riemann Hypothesis (RH) was introduced in [7]. | background 1.00 | 0.04 | 0.03 | asserted 1.00 |
| s1 | It states that the RH holds if and only if the span of a particular sequence of functions $(h_k)_{k\geq2}$, denoted $\mathcal{N}$, is dense in the Hardy space $H^2$. | background 1.00 | 0.07 | 0.08 | asserted 1.00 |
| s2 | In this note we construct a bounded operator $T$ on $H^{2}$ such that if the closure of $\mathcal{N}$ is $T$-invariant, then either the RH holds or there exists an infinite progression of $\zeta$-zeros. | contribution 0.68 | 0.95 | 0.83 | asserted 0.99 |
| s3 | In particular, this yields an operator-theoretic criterion for the Riemann hypothesis as an invariant subspace question. | interpretation 0.99 | 0.82 | 0.35 | asserted 0.98 |
What was examined, and what was not
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References
Discussion
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