Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Number Theory and Special Functions
- Paper type
- Research Article
- Manuscript stage
- Preprint
We prove uniform-norm small-ball bounds of logarithmic order ε⁻² about every admissible 1/2-Hölder center of the Kloosterman limit process, including boundary coefficient and endpoint values. Graded Fejér approximation removes two logarithmic losses from an inspected earlier lower estimate. We determine worst-case power orders over admissible Hölder classes, give an explicit O(log log p/√log p) uniform-path Wasserstein convergence rate for prime Kloosterman paths with both parameters averaged, and derive uniform shrinking-target counts. Extensions to bounded symmetric unimodal coefficient laws include sharp noncritical boundary power thresholds and a fixed-center obstruction at equality. The critical logarithmic gap remains unresolved. Classical inputs and inspected prior work are credited; no priority or optimal-rate claim is made.
Edition 2
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Number Theory and Special Functions
- Paper type
- Research Article
- Manuscript stage
- Preprint
We classify the least degree of a polynomial whose square has negative signed-moment value for the normalized fixed-center Riemann theta product. On 3 ≤ x ≤ 12 the degree has fourteen transitions, takes values between five and fifteen, and is nonmonotone; every transition and equality case is determined. Jacobi symmetry extends the classification to π²/12 ≤ x ≤ 12. Two- and three-summand theta models approximate all fourteen crossings with respective root errors below 0.00049 and 2.588 × 10⁻¹⁰. Twenty explicit rational Gram certificates include a seven-point example stable under absolute entrywise errors of 1.94 × 10⁻⁷. The paper also establishes fixed-degree Gaussian limits and single-summand degree-five positivity.
Edition 2
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Number Theory and Special Functions
- Paper type
- Research Article
- Manuscript stage
- Preprint
We give a reproducible method for locating and certifying quintic obstructions for the normalized fixed-center Riemann theta product, and prove a unique transition in its odd quintic moment block on [3.64, 3.66]. The transition parameter lies strictly between 3.64888758188562 and 3.64888758188563. Within this interval, below the transition, degree five is the minimum degree of a polynomial whose square has negative signed-moment value; at the transition the degree-five form is positive semidefinite with a one-dimensional nullspace; above it the form is positive definite. The proof combines a Schur-complement reduction with uniform directed interval bounds, including the complete infinite theta tails. An exact rational arithmetic replay verifies the bounds on the entire interval. An explicit integer-coefficient quintic illustrates why testing one fixed polynomial does not locate the transition of the full quintic block.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Number Theory and Special Functions
- Paper type
- Replication and Reproducibility Study
- Manuscript stage
- Preprint
We give a complete rational reconstruction of an explicit obstruction to positive definiteness for a fixed-center product of the Riemann theta density. At s₀ = ¼ log(13/(4π)), the signed derivative functional evaluated on the square of P(X) = X⁵ − 264X³ + 10952X lies in the exact rational interval [−1412370791.837791749, −1412370791.837791748]. The proof constructs derivatives by formal power series, retains twelve theta summands, bounds the entire remaining series, and encloses exponentials by range-reduced alternating series. A printed table of rational moment intervals proves the negative contraction. The odd derivative Hankel block is indefinite, and the six-point translation Gram has a negative direction at every sufficiently small positive spacing.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Number Theory and Special Functions
- Paper type
- Research Article
- Manuscript stage
- Preprint
We exhibit an explicit polynomial square on which the signed derivative functional of a fixed-center Riemann theta product is negative. At the center s₀ = ¼ log(13/(4π)), the polynomial P(X) = X⁵ − 264X³ + 10952X satisfies ℒₛ₀(P²) ≤ −1,351,625,671. The certificate uses rational interval arithmetic for derivatives through order ten and an explicit bound on the complete infinite theta tail. It follows that the normalized product at this center is not positive definite and that its odd derivative Hankel block is indefinite.