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math.CV — Complex Variables

Origin-Centered Möbius Circles and Dyadic Pairings of the Riemann Zeta Function on the Critical Line

Contributed by Ryan Hackbarth

We investigate the complex geometry of the normalized Dirichlet eta terms tn(s) = 1 1−2 1−s (−1)n−1 ns (s = a + bi). By analyzing the Möbius geometry of each term, we demonstrate that the base-2 resonant partial sum S2(s) = t1(s) + t2(s) forms an exact circle. Remarkably, this base circle is centered strictly at the origin (0, 0) if and only if Re(s) = 1 2 . We further establish a dyadic pairing theorem showing that every oddeven pair Pm(s) = tm(s) + t2m(s) inherits this origin-centered circular symmetry on the critical line. Analyzing the phase distribution on these concentric circles reveals four geometric mechanisms that distinguish non-trivial Riemann zeta zeros b = bn from generic values of b, including right-half-plane slow-zone clustering of P1 and opposinghemisphere alignment of the leading inner circles P3, P5, P7. Finally, we prove that removing all odd-even dyadic pairs leaves the multiples-of-4 subseries P∞ k=1 t4k(s) = t4(s)ζ(s), whose partial sums spiral around a target center Ctarget(b) that coincides with the origin (0, 0) if and only if b is a Riemann zeta zero.

A mix of human-written and AI-generated textHuman understanding: all parts

Oct 23, 2026PDF ↗View work →

hexagon:2610.00486v1

math.PR — Probability

Explicit onsets, rates and the tail of the collision constant for heat-evolved Weyl polynomials

Contributed by Bingo Baggins

Let $h_k$ be the orthonormal Hermite polynomials for the standard Gaussian weight, and let $a(w)=\sum_{j\ge0}a_jw^j$ be a real power series with radius of convergence greater than $1$ , $a(0)\ne0$ , no zeros on the unit circle and only simple zeros in the unit disk. For all large $n$ the polynomial $\sum_{j=0}^{n}a_jh_{n-j}$ has exactly as many nonreal zeros as $a$ has in the disk, but the known proof gives the onset by compactness. We make it explicit: if $a$ has no zeros in $\{1-d\le|w|\le1\}$ , with $d$ below an explicit threshold, the count is exact as soon as $n$ exceeds a polynomial in $1/d$ and in the inverses of $|a(0)|$ , $\min_{|w|=1}|a|$ , $\min_{|w|=1-d}|a|$ and $\min_{[-1+d,1-d]}\max(|a|,|a'|)$ ; no separation of the nonreal zeros of $a$ is needed. By an exact identity this applies to the random Weyl polynomials $G_n$ with independent standard Gaussian coefficients under the backward heat flow $G_{n,t}=\exp(-\frac t{2n}\partial_z^2)G_n$ , whose real-rooting time converges in distribution to $T_*=R_*^{-2}$ , where $R_*$ is the smallest modulus of a nonreal zero of the Gaussian power series $q(w)=\sum_j\eta_jw^j$ . We show that the convergence holds with the rate $n^{-\alpha}$ on $[1+\varepsilon,\infty)$ for every $\alpha<\frac3{50}$ , an exponent that does not depend on $\varepsilon$ ; the dependence on $\varepsilon$ is removed by a super-exponential tail for the number of zeros of $q$ in a disk. Under a coupling of the degrees that shares the top coefficients, the real-rooting time $\Lambda_n$ satisfies $|\Lambda_n-T_*|\le K/n$ for $n\ge n_0$ , with $K$ and $n_0$ depending on $q$ . We bound $K$ and $n_0$ by polynomials in condition parameters of $q$ , with absolute degrees and, when $T_*\ge1+\varepsilon$ , coefficients polynomial in $1/\varepsilon$ . For the Gaussian power series this gives, on the event $\{T_*\ge1+\varepsilon\}$ , a power tail with exponent $1/900$ for the largest scaled displacement $D$ of the final collision of zeros, and without restriction the tail bound $\exp(-c(\log x/\log\log x)^{1/2})$ ; the inputs include closed forms of the two-point correlation functions of the zeros of $q$ .

Primarily AI-generated textHuman understanding: no partsHermite polynomialsWeyl polynomialscollision of zeroscorrelation functions of zerosexplicit onsetheat flowlinear combinations of orthogonal polynomialsnonreal zerosrandom power seriesrate of convergence

Oct 23, 2026PDF ↗View work →

hexagon:2610.00485v1

math.NT — Number Theory

A 256-step certificate for Turán's pure power-sum constant

Contributed by Ricardo Pinho

We give an exact-input certificate for the upper bound C_42 <= 0.6888863712 in the pure power-sum problem. The construction refines the step profiles of Griego, Röhrig and Plata to 256 middle blocks, without changing their coefficient-cancellation mechanism. It improves the strongest public proposal located in our 22 October 2026 audit, 0.688970, by 0.0000836288. An unmodified public interval checker and a separately written Arb checker verify the strict finite inequalities. The asymptotic implication is proved in the note; no explicit finite threshold, optimality, human peer review or formal verification is claimed.

Primarily AI-generated textHuman understanding: some partsTurán power sumscoefficient cancellationcomputer-assisted proofinterval arithmeticrigorous numerics

Oct 23, 2026PDF ↗View work →

hexagon:2610.00484v1

math.MG — Metric Geometry

A counterexample to the table conjecture on the sphere

Contributed by Pablo Soberón

We disprove Matschke's table conjecture, which asserts that every continuous real-valued function on $S^2$ takes a common value on the vertices of some rotated copy of every prescribed inscribed square. The counterexample is the cubic polynomial $H(x,y,z)=14z^3-6z+y+xz$ and a square contained in a plane at distance $1/\sqrt{7}$ from the origin. We give an elementary proof.

Primarily human-written textHuman understanding: all partscounterexamplefixed pointsquare-peg

Oct 23, 2026PDF ↗View work →

hexagon:2610.00483v1

math.PR — Probability

The Real-Rooting Time Of Heat-Evolved Random Polynomials

Contributed by Bingo Baggins

Let $G_{n,t}=\exp(-\frac{t}{2n}\partial_z^2)G_n$ be a random Weyl polynomial of degree $n$ under the backward heat flow, with independent coefficients of a law that is not a point mass and has finite variance. Hall, Ho, Jalowy and Kabluchko showed that the empirical zero distribution collapses onto the real line at $t=1$ . We show that the first time at which all zeros of $G_{n,t}$ are real, a finite-degree analogue of the de Bruijn--Newman constant, converges in distribution to $T_*=R_*^{-2}$ , where $R_*$ is the smallest modulus of a nonreal zero of the random power series $\sum_j\eta_jw^j$ with coefficients of the same law; the convergence is almost sure under a coupling of the degrees that shares the top coefficients, whereas under the standard coupling the set of limit points of the real-rooting time is almost surely $[1,\infty]$ . The proof rests on an exact count of the nonreal zeros of analytic combinations $\sum_{j\le n}a_jh_{n-j}$ of orthonormal Hermite polynomials: if the real power series $a$ has radius of convergence greater than $1$ , no zeros on the unit circle and simple zeros in the unit disk, then for all large $n$ the combination has exactly as many nonreal zeros as $a$ has in the disk. Its method expresses the polynomial and its derivative in the bulk through a $2\times2$ matrix applied to $(h_n,h_{n-1})$ , built from only $O(\log n)$ steps of the backward three-term recurrence, with the remaining terms controlled by a Christoffel--Darboux bound. For Gaussian coefficients the law of $T_*$ is a Fredholm determinant, $\mathbb{P}(T_*>t)\sim\frac4{3\pi}t^{-3/2}$ as $t\to\infty$ , and $(t-1)\log\mathbb{P}(T_*\le t)\to-\pi^2/12$ as $t\downarrow1$ , half of the constant for complex coefficients. At the scale $1/n$ the last pair of nonreal zeros collides inside the real bulk, at relative position $\cos\theta_*$ , where $R_*e^{\pm i\theta_*}$ are the nonreal zeros of minimal modulus, and the real-rooting time is within $O(1/n)$ of $T_*$ . In a window of size $1/n$ the polynomial is, up to a positive factor and $O(1/n)$ , the model $\zeta\cos(\zeta+c)+\lambda\sin(\zeta+c)$ , whose threshold for real-rootedness is given by Kepler's equation; the phase $c$ rotates with $n$ , so $n(\Lambda_n-T_*)$ oscillates over an explicit interval and has an explicit limit law jointly with $T_*$ .

Primarily AI-generated textHuman understanding: no partsFredholm determinantsHermite polynomialsPfaffian point processesWeyl polynomialscollision of zerosde Bruijn-Newman constantheat flowlinear combinations of orthogonal polynomialsrandom power seriesreal zeros