Number Theory

2609 Submissions

[4] viXra:2609.0027 [pdf] submitted on 2026-09-11 20:47:48

Theta Geometry and Hankel Positivity in the Riemann Hypothesis

Authors: Agum Abdillah
Comments: 40 Pages. (Note by viXra Admin: Please cite and list scientific reference and submit article written with AI assistance to ai.viXra.org)

This work develops a theta-geometric framework for studying the Hankel positivity problem associated with the Riemann Hypothesis.Starting from the centered Riemann xi-function, the framework exploits the even spectral symmetry to introduce a folded spectral coordinate and a meromorphic resolvent whose poles encode the squared centered zeros. Passing to reciprocal spectral variables leads to a canonical moment sequence whose Stieltjes moment property is characterized by the positivity of two associated families of Hankel matrices.The framework then connects this spectral-moment structure with the theta-function representation of the xi-function. A tilted theta measure is introduced, yielding an exact expectation representation for the folded resolvent and a differential hierarchy governing its derivatives. The resulting analysis leads to an adjoint differential operator and a two-variable kernel representation, together with an exact inhomogeneous differential identity in which boundary contributions are retained explicitly.The central outcome is an exact reduction of the spectral question to an all-orders Hankel positivity problem. The work distinguishes rigorously established identities and equivalences from numerical diagnostics and does not claim a proof of the Riemann Hypothesis. Instead, it provides a structured theta-geometric and moment-theoretic framework for investigating the remaining positivity problem.
Category: Number Theory

[3] viXra:2609.0024 [pdf] submitted on 2026-09-11 20:36:06

Obtaining Exponents in Fermat, Markov and Catalan Diophantine Equations

Authors: Pawel Jan Piskorz
Comments: 4 Pages.

We propose a procedure which allows to determine the only admissible exponents in Diophantineequations of three kinds: Fermat, Markov and Catalan. First, we use a set of rewriting rules for each Diophantine equation. Second, we obtain the parametrized versions of the Diophantine equations. Next, we differentiate each of parametrized equations obtaining the identities for Fermat, Markov and Catalan equations.
Category: Number Theory

[2] viXra:2609.0022 [pdf] submitted on 2026-09-08 06:47:50

The Unity of the Prime Number Theorem, Bertrand's Postulate and Euler's Proof

Authors: Yong Zhao
Comments: 4 Pages.

For any integer n greater than 1 such that 2n+1 is prime, we construct interval (n,2n). By synthesizing the prime number theorem, the Bertrand—Chebyshev theorem and Euler's proof to analyze the gap between 2n+1 and primes in the interval (n,2n), we prove the Polignac's Conjecture.For any integer greater than 1, we construct intervals (n,2n) and (2n,4n). By synthesizing the Bertrand—Chebyshev theorem and the prime number theorem to analyze the gap of primes between the consecutive intervals, we prove the Polignac's Conjecture.
Category: Number Theory

[1] viXra:2609.0016 [pdf] submitted on 2026-09-07 18:15:38

Twin Prime Formation Through a Proper Alliance of Layers of Odd Composite Sequences

Authors: Min Min Oo
Comments: 19 Pages. (Note by viXra Admin: Please cite and listed scientific references!)

This paper develops a Prime Pair Framework for studying twin primes using structured sequences of odd composites. The base sequenceO_1=9+6r_1, r_1≠((1+2x)^2-13)/6 " or" ((1+2x)^2-7)/6 provides intervals containing two odd integers, which may form twin primes if both are prime. When the general sequencesO_a=(2a+1)^2+(2+4a) r_a (a={2ⓜ,3ⓜ,4ⓜ,u2026ⓜ,k})and O_1are in proper alliance, the conditions are formed through lower, upper and double twin prime formation mechanisms.
Category: Number Theory