Curvature-Level Crossings of Riemann Zeta Argand Curves: Sharp Tails, Logarithmic-Depth Identity Control, and Named-Level Reductions
Year: 2026
Type: article
Abstract: This paper develops a geometric and probabilistic theory of the plane curves traced by the Riemann zeta function along vertical lines in the universality strip, where the real part lies strictly between one half and one. It studies how frequently these curves cross prescribed signed-curvature levels, how their large-curvature behavior reflects nearby critical points, and when statistical laws from a random Euler-product model transfer to the zeta function itself. The analysis connects differential geometry, analytic number theory, random analytic functions, singular derivative moments, and weighted combinatorial matching. A central geometric construction replaces the signed-curvature crossing equation by a denominator-free expression involving the first two derivatives of the zeta function. This cancellation produces a Kac–Rice crossing flux without negative powers of the first derivative, allowing the crossing problem to be studied through smooth joint distributions of analytic derivatives. For the random Euler product, the paper establishes smooth exponentially tilted logarithmic-derivative densities and proves that the crossing intensity is finite and positive. It also obtains a sharp large-curvature asymptotic: the intensity decays as the inverse square root of the curvature level, with the same leading coefficient for positive and negative levels and an error of inverse-first-power order. For the Riemann zeta function, the paper proves convergence of the normalized crossing count at every prescribed vertical line and every prescribed real curvature level. The limiting density agrees with the random-model intensity. The argument combines a reflected holomorphic construction, quantitative control of simultaneous small values, and a Jensen-based passage from analytic-function convergence to zero counts. Related results treat averaging over vertical-line locations and curvature levels, joint parameter averages, maximal windows, and recovery through shrinking parameter ranges. The work also establishes convergence of every fixed positive moment of the relevant negative logarithms at prescribed parameter pairs. Quantitative moment bounds distinguish local blocks from integrated strips and yield a stretched-exponential integrated tail estimate. These results control logarithmic collapse in the crossing problem while keeping separate the stronger singular estimates required for inverse derivative weights at critical points. A second major component develops finite-window curvature tomography. In this setting, each simple zero of the first derivative contributes a weight equal to the inverse square root of the magnitude of the second derivative. This relates extreme curvature to a weighted critical-point distribution. At growing height, the paper proves transfer for bounded marks, capped inverse-mark identities, and a precise fourth-power law for small derivative marks on logarithmically resolved ranges, retaining analytic multiplicities. It identifies an explicit weighted tail condition equivalent to removing the remaining singular cutoff. Target averaging provides a setting in which complete removal of that cutoff can be proved. For both the shifted first-derivative equation and the shifted second-derivative equation, the paper establishes convergence of inverse derivative-mark sums in target-space Lp norms whenever the inverse-mark exponent a and norm exponent p satisfy 0 < a ≤ 2 and 1 ≤ p ≤ 2/a. In particular, inverse-square-root marks reach the L4 endpoint. The limiting weighted target densities are smooth, the endpoint small-mark estimate admits a sharper logarithmic bound, and averages over shrinking target disks recover the limiting density at the disk center under an explicit relation between the radius and the norm error. Symmetry of the disk gives a quadratic averaging bias. These conclusions concern target-space convergence and averaging; they do not evaluate the singular finite-height sum at an individual exceptional target. The critical-point analysis further separates derivative marks into local cluster factors and zero-free background factors. On the explicit real-part interval from 0.99 to 0.993, displaced-disk Poisson–Jensen estimates and the classical eighth moment yield an inverse background moment exceeding 7/5 and a positive eighth background moment. These bounds control isolated roots and expanding families satisfying a shrinking cluster-factor cutoff, while isolating the unresolved contribution from exceptionally small cluster products. For suitably isolated two-root packets, the paper constructs an injective correspondence with simple zeros of the second derivative. The normalized critical value at the image point is comparable to the square of the original root separation. The corresponding random weighted critical-value law has a fourth-power leading term and an eighth-power remainder. Its fixed-scale arithmetic counterpart follows by marked transfer, and a weighted estimate is proved at logarithmically shrinking scales. Extending this control to polynomially shrinking scales is reduced to a joint small-value estimate for the first two derivative coordinates. The actual Dirichlet coefficients are retained in the attack on that estimate. The paper proves power-saving twisted quadratic and fixed even-moment bounds and gives an exact cofactor decomposition obtained by conditioning on prime phases above the square-root cutoff. On a quantitatively specified cofactor family, this decomposition yields a nonlinear conditional Haar density bound for the derivative pair. A complementary argument shows that replacing every conditional distribution by a uniform density majorant incurs a cost incompatible with the existing mean-square approximation budget on shrinking scales. This identifies a limitation of that particular method and motivates estimates preserving the dependence between the actual remainder and its cofactor vectors. Conditional Haar estimates are explicitly distinguished from estimates along the arithmetic vertical orbit. Weighted Hall certificates extend the cluster analysis beyond isolated pairs. For polynomials with simple roots at every finite degree, and for local holomorphic packets satisfying explicit background and boundary conditions, the paper assigns inverse derivative demands at roots to capacities at critical points. Critical-point multiplicities are retained, including merged critical points. The construction uses nested lemniscate components, critical-point counting, and Hall matching. It provides explicit cut reductions for overlapping path and cycle families and a sufficient resource-moment condition for a vanishing singular root tail. Exact examples and reproducible computations accompany these arguments, including rational demand and capacity bounds, an explicit cubic matching certificate, a three-root example in which every proper cut passes but the total cut fails, and 300 rational path-and-cycle verification cases. Analytic and arithmetic countermodels clarify why fixed-scale convergence, bounded observables, or separate fixed-order estimates cannot automatically supply the required singular-tail control. The remaining open problems are stated precisely: the polynomial-scale vertical small-value estimate for the actual coefficients, arithmetic control of the Hall capacity tails with the necessary cluster-size and overlap uniformity, and complete inverse-square-root uncapping at the original zero target. The paper therefore combines proved curvature-crossing and target-averaged limit theorems with explicit reductions and certificates for the remaining critical-mark problem. Finite computations verify auxiliary identities and certificates; the analytic limit statements rest on the written proofs.
Author: Fathi, Kevin
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204584045
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