Functions and Analysis

2606 Submissions

[6] viXra:2606.0075 [pdf] submitted on 2026-06-20 21:33:52

A Dual Extension for the Gamma Function and the Confinement of the Zeros of the Riemann Zeta Function

Authors: Julinho Jorge Luís
Comments: 14 Pages.

The Bohr-Mollerup Theorem establishes the uniqueness of the Gamma function on the right half-plane, but makes no assertion regarding extensions to the left half-plane that preserve the integral form. This work proposes a Dual Architecture consisting of two integral operators with complementary domains: the Classical Gamma and a Symmetric Factorial, connected by an operator derived from the Hankel contour integral. When applied to the Riemann zeta function, this architecture replaces the classical functional equation—which exhibits indeterminate forms at integer points—with a formulation that is directly evaluable and preserves all values of the Dirichlet series. Analysis of the connection operator reveals that its real part vanishes exclusively on the critical line within the critical strip. Assuming a non-trivial zero off this line leads, via a closed cycle of the dual functional equation, to a contradiction involving the modulus of the Gamma function, as established independently through the Weierstrass representation and the maximum modulus principle. The result forces all non-trivial zeros of the Riemann zeta function to lie on the critical line.Keywords: Gamma function, Riemann zeta function, critical line, Hankel contour, Weierstrass representation.
Category: Functions and Analysis

[5] viXra:2606.0074 [pdf] submitted on 2026-06-20 22:13:49

Arbitrary Approximation of the Shifted-LRC by the LRC

Authors: Deepak Ponvel Chermakani
Comments: 2 pages, 2 Theorems.

Consider an instance of the Shifted Lonely Runner Conjecture (S-LRC) where all n runners (except the stationary runner 0) have integer speeds and start from real values in [0,1[ at time t=0. We show that one can derive an alternative vector of starting points that can be made to be arbitrarily close to the initial vector of starting points. The alternative starting point of each runner i is a rational in [0,1[ and is expressible as (qi / P) where P is a large prime and qi is an integer in [0, P-1]. The S-LRC instance with the alternative starting points, allows a minimal loneliness gap of f, if and only if, the corresponding LRC allows a minimal loneliness gap of f, where f is a desired fraction in ]0,1[. This finding is important in the light of recent counter-examples to the shifted-LRC.
Category: Functions and Analysis

[4] viXra:2606.0053 [pdf] submitted on 2026-06-14 21:00:26

An Optimized Sech-Based Approximation of E^{-X^2}

Authors: Vladyslav Vasilache
Comments: 2 Pages. (Note by viXra Admin: Please cite and list scientific reference and submit article written with AI assistance to ai.viXra.org)

This paper introduces a new, highly accurate approximation for the function $e^{-x^2}$. By differentiating a known error function approximation and optimizing its parameters, we drastically reduce the maximum absolute error from $1.88%$ to less than $0.09%$ without using any exponential terms.
Category: Functions and Analysis

[3] viXra:2606.0035 [pdf] submitted on 2026-06-10 10:49:40

Elementary Weak Solutions of the Navier—Stokes Equations and an Application of the Banach Fixed-Point Theorem

Authors: Masatoshi Ohrui
Comments: 23 Pages.

This is an application of functional analysis to the existence and smoothness of the Navier—Stokes equations using elementary weak solutions in Sobolev spaces.We solve the problem in mathematics. The problems are not in physics, so we do not use any physics or assumptions-falsified mathematics, such as other papers. We use mathematics only. We can solve the problem by using an exactly and completely FALSIFIED resolution, where large initial values destroy the earth, because uniqueness does NOT hold, or SMALL initial values love your cup of coffee.There are no long or complicated calculations; semi-groups, a priori estimates, and boundary conditions are not used at all. We apply the local solvability of linear partial differential operators with constant coeficients.
Category: Functions and Analysis

[2] viXra:2606.0034 [pdf] submitted on 2026-06-10 10:55:32

Proof of Hartog’s Phenomenon and Cohomology Vanishing Theorem

Authors: Masatoshi Ohrui
Comments: 2 Pages.

We can prove Hartog’s phenomenon by solving the ∂-bar equation for compactly supported forms. To solve the equation, we construct the solution using convolution.
Category: Functions and Analysis

[1] viXra:2606.0017 [pdf] submitted on 2026-06-06 18:49:02

Hastings-Cody Approximations of the Integral of a Power Times the Complete Elliptic Integral of the First Kind

Authors: Richard J. Mathar
Comments: 16 Pages.

Hastings and later Cody tabulated minimax polynomial approximations for the Complete Elliptic Integral of the First Kind. The simplicity of this representation by polynomials and polynomials times a logarithm allows to integrate their terms analytically. We demonstrate how integrals of the Complete Elliptic Integral times a power of its argument achieve double precision accuracy for powers from 0 to 2 based on Cody's polynomials up to 9th order.
Category: Functions and Analysis