REVIEW
A Hardy space approximation supporting zero-free half-planes for the $\zeta$-function
Functional Analysis · Submitted 2 Nov 2026 · 20 references
Pith's verdict: Solid theoretical extension of Noor's convergence to half-planes, plus careful numerics with honest limitations.
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Referee report
The paper
The paper claims
The paper shows that the Möbius expansion of $1/s$ converges in the Hardy spaces of the half-planes $\mathrm{Re}(s)>\alpha$ for every $\alpha>1$, and that a single zero of $\zeta$ in such a half-plane keeps the best approximation error…
The reading
What carries the argument
Two devices carry the argument. The first is the transfer operator $M_\tau$, defined by $(M_\tau f)(s)=f(s-\tau)/s$, which is bounded from the Hilbert space of Dirichlet series with square-summable coefficients into $H^2(\mathbb{C}_{1/2+\tau})$ and sends the constant function $1$ to $E(s)=1/s$ and $f_{k,\tau}(s)=(1-k^{1-s-\tau})\zeta(s+\tau)$ to $-kG_k(s)$; this is what turns a disk-space statement about $\sum_k\mu(k)k^{-\tau}g_{k,\tau}$ into a half-plane statement about the Möbius partial sums. The second is the reproducing kernel $k_w(s)=1/(s+\bar w-2\alpha)$ of $H^2(\mathbb{C}_\alpha)$: since each $G_k$ vanishes at every zero of $\zeta$, the kernel at a zero pins the value $1/\rho$ onto any candidate approximation, and Cauchy–Schwarz converts that single pinned value into a floor on the whole error. Feeding the same kernel identity into the Mellin picture yields the zeta-free error series of Proposition 4.1, whose non-negative terms make the $n\le10^6$ computation a plain summation rather than a delicate cancellation.
Extended reading notes
Core claim
The central claim is a two-sided sandwich on a single threshold. On one side, Lemma 3.1 transfers Noor's disk-space convergence to the family $g_{k,\tau}(z)=k^{\tau-1}\sum_n z^n/n^\tau-\sum_n z^{nk}/n^\tau$, showing $\sum_k \mu(k)k^{-\tau}g_{k,\tau}\to -z$ in $H^2_0$ for each $\tau>1/2$, and a bounded operator $M_\tau:f\mapsto f(s-\tau)/s$ from the Hilbert space of Dirichlet series into $H^2(\mathbb{C}_{1/2+\tau})$ carries this to $S_n=\sum_{k=2}^n\mu(k)G_k\to E$ in $H^2(\mathbb{C}_\alpha)$ for every $\alpha>1$ (Theorem 3.2, Corollary 3.3). On the other side, because $G_k(\rho)=0$ at every zero of $\zeta$ and the space is a reproducing-kernel space, the evaluation functional at a zero $\rho$ gives $\|E-\sum c_kG_k\|_\alpha^2\ge |E(\rho)|^2/\|k_\rho\|_\alpha^2=2(\mathrm{Re}\,\rho-\alpha)/|\rho|^2$ for all coefficients, with a determinant form $v^*K^{-1}v$ for several zeros (Theorem 3.5). Hence the threshold $\alpha_c=\inf\{\alpha\ge1/2: d_n(\alpha)^2\to0\}$ satisfies $1/2\le\Theta\le\alpha_c\le1$, where $\Theta$ is the supremum of the real parts of the zeros, and $\alpha_c=1/2$ exactly under RH (Corollary 3.6). The numerical half rests on Proposition 4.1, which writes $s_n(\alpha)=\|S_n-E\|_\alpha^2$ as $\sum_{m\ge1}w_m(\alpha)R_n(m)^2$ with $w_m(\alpha)=(m^{-2\alpha}-(m+1)^{-2\alpha})/(2\alpha)$ and $R_n(m)=1+\sum_{k=2}^n\mu(k)(m\bmod k)/k$; computing this to $n=10^6$ shows $s_n(1/2)$ oscillating without decay, while for $\alpha>1/2$ the envelope falls like $n^{1-2\alpha}$, far faster than the conditional bound $n^{-(\alpha-1/2)/3}$ available under RH, and on the critical line the optimal error satisfies $d_n(1/2)^2\log n\approx0.046$ for $100\le n\le5120$, matching the constant $2+\gamma-\log4\pi$ and earlier computations.
Load-bearing premise
Beyond the proved range $\alpha>1$, everything the paper asserts about the strip rests on reading a power law $s_n(\alpha)\asymp n^{1-2\alpha}$ off the envelope of a sequence whose values swing by factors of hundreds within one decade and which is sampled at 3001 points per decade, and the authors themselves note that a zero just off the line would produce a floor of order $10^{-24}$ that no finite computation could see.
Editorial extensions
If this is right
- For every $\alpha>1$ the Möbius expansion of $1/s$ is not merely formal: the partial sums $\sum_{k=2}^n\mu(k)G_k$ converge in $H^2(\mathbb{C}_\alpha)$, which already delivers the zero-free half-plane $\mathrm{Re}(s)>1$ by the reproducing-kernel argument.
- The failure Báez-Duarte found on the critical line is the endpoint of a scale: for each $\alpha\ge1/2$, convergence of the expansion in $H^2(\mathbb{C}_\alpha)$ implies $\zeta$ has no zeros with real part exceeding $\alpha$, so any proof of convergence at some $\alpha<1$ would produce a zero-free half-plane beyond what is currently known.
- A single zero $\rho$ with $\mathrm{Re}(\rho)>\alpha$ gives the explicit floor $d_n(\alpha)^2\ge 2(\mathrm{Re}\,\rho-\alpha)/|\rho|^2$ for every $n$, and several zeros give $v^*K^{-1}v$ with $K_{ij}=(\rho_i+\bar\rho_j-2\alpha)^{-1}$; the optimal error therefore cannot tend to zero in any half-plane containing a zero.
- The threshold $\alpha_c$ where the optimal error vanishes satisfies $1/2\le\Theta\le\alpha_c\le1$ and equals $1/2$ if and only if RH holds, so RH is equivalent to a statement about where a single numerical sequence starts decaying.
- The error $s_n(\alpha)$ has an exact series representation with elementary terms and no zeta function, so the entire strip can be tabulated; the tables show $s_n(1/2)$ oscillating between roughly $3\times10^{-2}$ and $3\times10^{-1}$ while $s_n(\alpha)$ for $\alpha>1/2$ decays close to $n^{1-2\alpha}$.
Reading between the lines
- If the observed $n^{1-2\alpha}$ envelope is genuine rather than a transient over $10^3\le n\le10^6$, then the partial sums $M_1(n)=\sum_{k\le n}\mu(k)/k$ must reach size of order $n^{-1/2}$ along a sequence of $n$, since the authors' own lower bound (4.2) shows the decay cannot outrun $M_1(n)^2 n^{2-2\alpha}$; the numerics are therefore implicitly a statement about the size of Mertens-type sums.
- The gap between the proved region $\alpha>1$ and the numerically probed region $\alpha>1/2$ is exactly the gap between the prime number theorem and RH; pushing Lemma 3.1 to $\tau=1/2$ would be a direct route to the critical line, and the divisor-count estimate used there is the visible place where the method refuses to go.
- Because $d_n(1/2)^2$ is trapped between Burnol's unconditional $\liminf$ bound with constant $2+\gamma-\log4\pi$ and the computed values hovering within a few per cent of it, an independent computation at substantially larger $n$ would discriminate between a genuine asymptotic and a slow crossover, and is cheap compared with proving anything.
- The reproducing-kernel floor is slack by design: $2(\mathrm{Re}\,\rho-\alpha)/|\rho|^2$ for a hypothetical zero near the critical line is around $10^{-24}$ for zeros of moderate height, which is why no finite computation of $s_n(\alpha)$ can exclude such a zero and why the numerical evidence is evidence about the coefficients, not about the zeros.
Original abstract
By a theorem of Bagchi, the Riemann Hypothesis (RH) holds if and only if $E(s) = 1/s$ belongs to the closed linear span of $G_k(s) = (k^{-s} - k^{-1})\zeta(s)/s$, $k \geq 2$, in the Hardy space $H^2(\mathbb{C}_{1/2})$, where $\mathbb{C}_\alpha$ is the half-plane $\mathrm{Re}(s) > \alpha$. The formal M\"obius expansion $E = \sum_{k \geq 2} \mu(k) G_k$ diverges on the critical line, by a theorem of B\'aez-Duarte. We study it in the scale of spaces $H^2(\mathbb{C}_\alpha)$, where its convergence forces $\zeta$ to be zero-free in $\mathbb{C}_\alpha$. We prove that the partial sums $S_n$ converge to $E$ in $H^2(\mathbb{C}_\alpha)$ for every $\alpha > 1$, and that a single zero of $\zeta$ in $\mathbb{C}_\alpha$ keeps the best-approximation error $d_n(\alpha)^2$ uniformly away from zero. We express the M\"obius error $s_n(\alpha) = \|S_n - E\|^2_\alpha$ as a series with elementary terms, free of $\zeta$, and compute it for $n \leq 10^6$: $s_n(1/2)$ oscillates without decaying, whereas for $\alpha > 1/2$ it decays at a rate close to $n^{1-2\alpha}$, much faster than the bound known under RH. On the critical line, $d_n(1/2)^2$ behaves like $0.046/\log n$ for $100 \leq n \leq 5120$, in agreement with earlier computations by B\'aez-Duarte, Balazard, Landreau and Saias.
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
By a theorem of Bagchi, the Riemann Hypothesis (RH) holds if and only if $E(s) = 1/s$ belongs to the closed linear span of $G_k(s) = (k^{-s} - k^{-1})\zeta(s)/s$, $k \geq 2$, in the Hardy space $H^2(\mathbb{C}_{1/2})$, where $\mathbb{C}_\alpha$ is the half-plane $\mathrm{Re}(s) > \alpha$. The formal M\"obius expansion $E = \sum_{k \geq 2} \mu(k) G_k$ diverges on the critical line, by a theorem of B\'aez-Duarte. We study it in the scale of spaces $H^2(\mathbb{C}_\alpha)$, where its convergence forces $\zeta$ to be zero-free in $\mathbb{C}_\alpha$. We prove that the partial sums $S_n$ converge to $E$ in $H^2(\mathbb{C}_\alpha)$ for every $\alpha > 1$, and that a single zero of $\zeta$ in $\mathbb{C}_\alpha$ keeps the best-approximation error $d_n(\alpha)^2$ uniformly away from zero. We express the M\"obius error $s_n(\alpha) = \|S_n - E\|^2_\alpha$ as a series with elementary terms, free of $\zeta$, and compute it for $n \leq 10^6$: $s_n(1/2)$ oscillates without decaying, whereas for $\alpha > 1/2$ it decays at a rate close to $n^{1-2\alpha}$, much faster than the bound known under RH. On the critical line, $d_n(1/2)^2$ behaves like $0.046/\log n$ for $100 \leq n \leq 5120$, in agreement with earlier computations by B\'aez-Duarte, Balazard, Landreau and Saias.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | By a theorem of Bagchi, the Riemann Hypothesis (RH) holds if and only if $E(s) = 1/s$ belongs to the closed linear span of $G_k(s) = (k^{-s} - k^{-1})\zeta(s)/s$, $k \geq 2$, in the Hardy space $H^2(\mathbb{C}_{1/2})$, where $\mathbb{C}_\alpha$ is the half-plane $\mathrm{Re}(s) > \alpha$. | background 1.00 | 0.04 | 0.08 | asserted 1.00 |
| s1 | The formal M\"obius expansion $E = \sum_{k \geq 2} \mu(k) G_k$ diverges on the critical line, by a theorem of B\'aez-Duarte. | background 0.98 | 0.03 | 0.03 | asserted 1.00 |
| s2 | We study it in the scale of spaces $H^2(\mathbb{C}_\alpha)$, where its convergence forces $\zeta$ to be zero-free in $\mathbb{C}_\alpha$. | contribution 0.89 | 0.37 | 0.16 | asserted 0.92 |
| s3 | We prove that the partial sums $S_n$ converge to $E$ in $H^2(\mathbb{C}_\alpha)$ for every $\alpha > 1$, and that a single zero of $\zeta$ in $\mathbb{C}_\alpha$ keeps the best-approximation error $d_n(\alpha)^2$ uniformly away from zero. | result 0.96 | 0.94 | 0.43 | asserted 1.00 |
| s4 | We express the M\"obius error $s_n(\alpha) = \|S_n - E\|^2_\alpha$ as a series with elementary terms, free of $\zeta$, and compute it for $n \leq 10^6$: $s_n(1/2)$ oscillates without decaying, whereas for $\alpha > 1/2$ it decays at a rate close to $n^{1-2\alpha}$, much faster than the bound known under RH. | result 0.95 | 0.94 | 0.34 | asserted 1.00 |
| s5 | On the critical line, $d_n(1/2)^2$ behaves like $0.046/\log n$ for $100 \leq n \leq 5120$, in agreement with earlier computations by B\'aez-Duarte, Balazard, Landreau and Saias. | result 1.00 | 0.51 | 0.09 | asserted 0.91 |
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