Let (a_k) be the positive coefficient sequence of the normalized Riemann xi-function, and let D_{r,k} denote its consecutive Toeplitz minors. The Riemann Hypothesis is equivalent to (a_k) being a Polya frequency sequence of infinite order, and hence to nonnegativity of all Toeplitz minors. We prove that D_{r,k}>0 for every r>= 2 and k>= 10^18 r^3. This gives an explicit cubic tail scale uniform in r, in contrast with Katkova's fixed-order asymptotic positivity. The proof does not use numerically verified zeros of the Riemann zeta-function. It combines a certified complex saddle-point analysis of the moment transform, an exact q-Pascal dilation semigroup controlling every degree simultaneously, and a weighted Banach-algebra majorant for the nonlinear remainder, closed by an inertia-preservation argument. All analytic constants are certified with directed rounding in Arb ball arithmetic, and all algebraic identities are verified in exact rational arithmetic. The ancillary files reproduce every certificate. The result concerns only the tail regime k much larger than r^3 and makes no progress on the Riemann Hypothesis, which concerns the complementary region.
Let $f_m(x_0,\ldots,x_{2m-2}) =\det(x_{i+j})_{0\leq i,j\leq m-1}$ be the determinant of the generic symmetric Hankel matrix of size $m$. We determine its global Bernstein--Sato polynomial. The proof combines the nearby cycles of the corresponding secant hypersurface with an explicit Archimedean zeta integral over the c...
Let D be the open unit disk and let R denote the class of normalized analytic functions whose derivatives have positive real part. We determine the sharp upper bound 8/125 for the absolute value of the fourth Hankel determinant H(4,1). The determinant is represented as an integer polynomial in the first six Carathéodor...
Musthafa Ibrahim, Zainab H. Mahmood, Reem O. Rasheed et al.· Basrah journal of science· 0 citations
We prove the strict positivity conjectured by Chenevier for the critical vectors in the unconditional part of his automorphic Hermite--Minkowski theorem. The proof establishes strict negativity of all Verblunsky coefficients of a circle measure associated with the weight $(\arcsin x)/x$ on $(-1,1)$. A positive-kernel f...
Let $(E,h)$ be a singular Hermitian vector bundle on a complex manifold $X$, and let \[ \mathcal E(h)_x=\{F\in\mathcal O(E)_x:|F|_h^2\in L^1_{\mathrm{loc},x}\} \] be its multiplier submodule sheaf. We introduce a finite plurisubharmonic Gram scalarization condition under which the higher-rank integrability problem redu...
Let X be a centered infinitely divisible random vector with finite second moment and covariance matrix Sigma. We define the radial pole C_X(t) as the smallest scale in a right sub-gamma bound forwhose quadratic proxy is fixed at the true variance t^T Sigma t. A canonical directional measure and an independent Beta(1,2)...
For fixed $k\ge2$, Garvan's modified $k$-rank moments are encoded by odd elliptic Taylor coefficients of an odd-level Appell function. The completed coefficient of order $2n+1$ has weight $2n+3/2$, so the family cannot be realized as a fixed-weight vector-valued modular form. Nevertheless, all non-holomorphic parts ari...
Seokho Jin, Sihun Jo· 0 citations
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