REVIEW
On the Sampling of Certain Dirichlet and Prime Sums at Gram Points and their Viscosity
Number Theory · Submitted 15 Oct 2026 · 23 references
Pith's verdict: The prime Gram cloud's two-lobe structure and its apparent viscosity ordering are genuinely new and worth checking, but the central correlation is never quantified and the AFE-confound worry from the stress test is not answered.
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The paper
The paper claims
Sampling the prime-indexed part of the Dirichlet polynomial at the Gram points of the Hardy Z-function yields a two-lobed geometric cloud whose internal structure is ordered by the local viscosity of the zeta function, encoding the…
The reading
What carries the argument
The load-bearing object is the approximate-functional-equation phase vector. At a Gram point the approximate functional equation expresses both $Z(g_n)$ and $Z'(g_n)$ through different projections of the same sum $E(g_n)=\sum_{k\le N(g_n)} e^{ig_n\log k}/\sqrt{k}$ — the unweighted real part for $Z(g_n)$, the logarithmically weighted imaginary part for $Z'(g_n)$ — so the complex value $E(g_n)$ contains precisely the phase information entering the viscosity quotient $\mu(g_n)=|Z'(g_n)/Z(g_n)|$. The paper filters this phase vector by multiplicative depth, writing $E(g_n)=1+\sum_{j\ge1}E_j(g_n)$ with $E_j$ the contribution of integers having exactly $j$ prime factors; the prime layer $E_1$ shows the strongest rank correlation with $\mu(g_n)$, and the correlation weakens as $j$ grows. The geometry is read through two lenses. The first is local dynamics: at an isolated bad Gram point with a confined pair of zeros — a length-two Gram block whose two zeros lie on the same side of $g_n$, a situation the dataset verifies via the two-zero rule with no violations among the first 250,000 Gram points — the zeros form a two-body system whose collision is detected by the Gram discriminant, and the viscosity measures the competition between the first-order collision tendency and the transverse escape. The second lens is probabilistic: treating the prime phases as weakly dependent angular variables, each lobe is described by the conditional mean and covariance of a weighted planar random walk, with nested convex-hull "onion layers" standing in for the level sets of the large-deviation rate function. Because the viscosity is exactly the imaginary part of the logarithmic derivative of $\zeta$ on the critical line, it is naturally controlled by the nearest zeros, which is what makes the correlation with a prime-phase sum plausible.
Extended reading notes
Core claim
Sampling at Gram points is far from neutral: the full cloud $\mathcal{C}_N=\{E(g_n)\}$ and the prime cloud $\mathcal{P}_N=\{E_1(g_n)\}$ form highly structured subsets of the ambient curves traced by $E(t)$ and $E_1(t)$ as $t$ varies. In the full cloud the good/bad separation is forced by the approximate functional equation, which gives $Z(g_n)=2(-1)^n\Re E(g_n)+O(g_n^{-1/4})$; in the prime cloud goodness is not defined, yet the isolated bad points still organize into two lobes above and below the good-point cloud. Folding the two lobes onto each other reveals a principal direction along which the viscosity $\mu(g_n)=|Z'(g_n)/Z(g_n)|$ varies systematically: the smallest viscosities cluster at one extremal end, near the conjectural repulsion boundary $\mu=4$, and the isolated-bad viscosity distribution has a sharp lower edge just above that value, with an observed minimum of about 4.26178. The lobe dichotomy is itself geometric: the upper lobe corresponds to the confined zero pair lying to the right of $g_n$, the lower lobe to the pair lying to the left, and the extremal low-viscosity points correspond to confined pairs in which one zero is displaced toward the outer wall $g_{n\pm1}$ of the local Gram interval. The data further suggest the normalized asymptotic lower limit $\liminf (2\pi/\log(g_n/2\pi))\,\mu(g_n)=2$ for isolated bad confined points. All of this happens at modest height, with $N(g_n)\le 164$ so the prime layer has at most 38 terms, a caution the paper itself flags against over-reading the rank correlations. The Davenport–Heilbronn analogue exhibits neither the viscosity bound, since its isolated-bad distribution has substantial mass below $\mu=4$ with a representative value of about 1.45, nor the two-lobe splitting, indicating that both phenomena are tied to the zero-protecting dynamics of the Hardy $Z$-function rather than to the formal functional equation alone.
Load-bearing premise
The load-bearing premise is that the good/bad classification and the underlying zero data are correct, and in particular that among the first 250,000 Gram points every isolated bad point obeys the empirical two-zero rule for its Gram strip, so that each such point really confines exactly two zeros; if any sampled point holds a different number of zeros, the two-lobe geometry and its viscosity ordering could be an artifact of the selected sample.
Editorial extensions
If this is right
- The principal-axis coordinate of the folded prime cloud acts as a visible ordering parameter for the viscosity, so a bad point's position in the lobe gives a first estimate of how close its local zero configuration sits to the repulsion boundary $\mu=4$.
- Smallest-viscosity isolated bad points acquire a concrete geometric meaning: they are the configurations in which one of the two confined zeros is pushed as far as possible toward the neighboring good Gram point, that is, toward the wall of the local Gram interval.
- The multiplicative-layer hierarchy shows that the local-zero information is carried mainly by the prime frequencies: layers built from integers with two or more prime factors lose the good/bad separation and the viscosity ordering progressively, with layers $E_4$ and $E_5$ essentially intermixed.
- The Davenport–Heilbronn control case links the absence of the viscosity bound with the absence of lobe splitting, implying that neither phenomenon follows from the functional-equation symmetry alone but reflects the specific collision-avoiding dynamics of the Hardy $Z$-function.
- Since the unweighted prime cloud $E_1(g_n)$ is a stripped projection of the weighted prime-power sum that the explicit formula attaches to the logarithmic derivative, the observed correlation suggests a simple arithmetic observable can proxy for local zero statistics; the paper notes that the explicit formula by itself does not explain why this survives in the simpler cloud.
Reading between the lines
- A direct out-of-sample extension the paper does not perform: recompute the folded prime cloud and the viscosity ordering for the next block of Gram points beyond the first 250,000, where the cited literature shows failures of the two-zero rule occur with positive proportion; the confinement-based reading predicts the two-lobe geometry and the $\mu$-ordering break exactly at the non-confined points
- The same construction applied to Dirichlet $L$-functions believed to satisfy the Riemann Hypothesis would test whether the lobe split and the boundary $\mu=4$ track the location of zeros rather than the specific arithmetic of $\zeta$; the paper runs only the one control case, the Davenport–Heilbronn function.
- Because the prime sum drops the logarithmic von Mangoldt weight and the prime powers that the explicit formula carries, re-weighting $E_1(g_n)$ and measuring whether the viscosity ordering sharpens or smears would quantify how much of the signal lives in the bare prime phases themselves.
- The lobe position of a bad point is effectively a cheap readout predicting on which side of $g_n$ its confined zero pair sits; verifying that prediction against direct zero locations in an independent height range would test the geometric encoding without invoking the deformation framework.
Original abstract
Consider the Dirichlet polynomial and its prime-indexed sub-sum \[ \begin{array}{ccc} E(t) := \sum_{k\le \left[ \sqrt{ \frac{t}{2\pi} } \right]} \frac{e^{it\log k}}{\sqrt{k}}, & ; & E_1(t) := \sum_{\substack{p\le \left[ \sqrt{\frac{t}{2\pi}} \right]\\ p\ {\rm prime}}} \frac{e^{it\log p}}{\sqrt p}. \end{array} \] In this work we investigate the statistical and arithmetic properties of the sets \[ \mathcal{C}_N := \left\{ E(g_n):1\le n\le N \right\}, \qquad \mathcal{P}_N := \left\{ E_1(g_n):1\le n\le N \right\}, \] obtained by sampling these sums at the Gram points \(g_n\) of the Hardy \(Z\)-function. We refer to \(\mathcal{C}_N\) and \(\mathcal{P}_N\) as the full and prime \(N\)-th Gram clouds, respectively. The motivation is that the approximate functional equation expresses both \(Z(g_n)\) and \(Z'(g_n)\) through different projections of the same oscillatory phase configuration, suggesting that the geometry of these sampled sums may retain information about the local behavior of nearby zeros. Numerically, we find that Gram-point sampling produces highly non-generic geometric structures: good and isolated bad Gram points occupy markedly different regions of the clouds, the prime cloud of isolated bad points splits into two anisotropic lobes, and the position along the principal direction of the folded prime cloud is strongly correlated with the viscosity of the Hardy $Z$-function \( \mu(g_n)=\left|\frac{Z'(g_n)}{Z(g_n)}\right|, \) with smallest-viscosity points concentrating near distinguished extremal regions of the lobes.
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
Consider the Dirichlet polynomial and its prime-indexed sub-sum \[ \begin{array}{ccc} E(t) := \sum_{k\le \left[ \sqrt{ \frac{t}{2\pi} } \right]} \frac{e^{it\log k}}{\sqrt{k}}, & ; & E_1(t) := \sum_{\substack{p\le \left[ \sqrt{\frac{t}{2\pi}} \right]\\ p\ {\rm prime}}} \frac{e^{it\log p}}{\sqrt p}. \end{array} \] In this work we investigate the statistical and arithmetic properties of the sets \[ \mathcal{C}_N := \left\{ E(g_n):1\le n\le N \right\}, \qquad \mathcal{P}_N := \left\{ E_1(g_n):1\le n\le N \right\}, \] obtained by sampling these sums at the Gram points \(g_n\) of the Hardy \(Z\)-function. We refer to \(\mathcal{C}_N\) and \(\mathcal{P}_N\) as the full and prime \(N\)-th Gram clouds, respectively. The motivation is that the approximate functional equation expresses both \(Z(g_n)\) and \(Z'(g_n)\) through different projections of the same oscillatory phase configuration, suggesting that the geometry of these sampled sums may retain information about the local behavior of nearby zeros. Numerically, we find that Gram-point sampling produces highly non-generic geometric structures: good and isolated bad Gram points occupy markedly different regions of the clouds, the prime cloud of isolated bad points splits into two anisotropic lobes, and the position along the principal direction of the folded prime cloud is strongly correlated with the viscosity of the Hardy $Z$-function \( \mu(g_n)=\left|\frac{Z'(g_n)}{Z(g_n)}\right|, \) with smallest-viscosity points concentrating near distinguished extremal regions of the lobes.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Consider the Dirichlet polynomial and its prime-indexed sub-sum \[ \begin{array}{ccc} E(t) := \sum_{k\le \left[ \sqrt{ \frac{t}{2\pi} } \right]} \frac{e^{it\log k}}{\sqrt{k}}, & ; & E_1(t) := \sum_{\substack{p\le \left[ \sqrt{\frac{t}{2\pi}} \right]\\ p\ {\rm prime}}} \frac{e^{it\log p}}{\sqrt p}. | background 0.64 | 0.16 | 0.02 | unclear 0.63 |
| s1 | \end{array} \] In this work we investigate the statistical and arithmetic properties of the sets \[ \mathcal{C}_N := \left\{ E(g_n):1\le n\le N \right\}, \qquad \mathcal{P}_N := \left\{ E_1(g_n):1\le n\le N \right\}, \] obtained by sampling these sums at the Gram points \(g_n\) of the Hardy \(Z\)-function. | contribution 0.87 | 0.71 | 0.04 | asserted 0.88 |
| s2 | We refer to \(\mathcal{C}_N\) and \(\mathcal{P}_N\) as the full and prime \(N\)-th Gram clouds, respectively. | contribution 0.44 | 0.40 | 0.03 | asserted 0.97 |
| s3 | The motivation is that the approximate functional equation expresses both \(Z(g_n)\) and \(Z'(g_n)\) through different projections of the same oscillatory phase configuration, suggesting that the geometry of these sampled sums may retain information about the local behavior of nearby zeros. | background 0.62 | 0.17 | 0.06 | hedged possibility 0.99 |
| s4 | Numerically, we find that Gram-point sampling produces highly non-generic geometric structures: good and isolated bad Gram points occupy markedly different regions of the clouds, the prime cloud of isolated bad points splits into two anisotropic lobes, and the position along the principal direction of the folded prime cloud is strongly correlated with the viscosity of the Hardy $Z$-function \( \mu(g_n)=\left|\frac{Z'(g_n)}{Z(g_n)}\right|, \) with smallest-viscosity points concentrating near distinguished extremal regions of the lobes. | result 1.00 | 0.95 | 0.84 | asserted 1.00 |
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Figures reproduced from arXiv:2610.21952 by the authors.
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