Functions and Analysis

2605 Submissions

[5] viXra:2605.0085 [pdf] submitted on 2026-05-20 22:14:22

A New Closed-Form Expansion Related to the Riemann Zeta Function

Authors: Ayoub Zaroual
Comments: 6 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

We establish a closed-form expansion of the Riemann zeta function ζ(s) at any complex point s = a + ib in terms of an auxiliary real-analytic function L(a) built from theBernoulli numbers and the rising factorial. The cornerstone of the derivation is a clean dif-ferentiation identity for the elementary symmetric polynomial function Kp n(a) on consecutive integers, which we prove by a generating-function argument. Specialising the expansion to the critical line Re(s) = 1/2 recasts the Riemann hypothesis as the simultaneous vanishing of two real power series in the imaginary part b.
Category: Functions and Analysis

[4] viXra:2605.0067 [pdf] submitted on 2026-05-16 20:45:59

Partial Differential Equations Arising From the Chain Rule for Direct Dependencies

Authors: Edigles Guedes
Comments: 3 Pages.

This paper presents two partial differential equations obtained from the chain rule for functions with direct dependencies among the variables.
Category: Functions and Analysis

[3] viXra:2605.0065 [pdf] submitted on 2026-05-16 20:36:42

Geometric Scaling Laws in Block Decompositions of the Dirichlet Eta Function

Authors: Min Min Oo
Comments: 9 Pages. (Note by viXra Admin: Author name is required in the article after the article title and the abstract should be labeled as such)

This paper presents a block-structured decomposition of the Dirichlet eta functionη(s)=∑_(n=1)^00 (-1)^(n-1) n^(-s), where s=σ+it, based on an exponential partitioning of the summation index into intervals of the form N_k∼2e^(2kπ/t). The series is reorganized into finite segments, which are analyzed as dynamically scaling components with approxipmately geometric decay behavior.A ratio structure between successive block contributions is derived, leading to an approximate exponential scaling law of the form R_k (s)∼e^(-2πσ/t). This allows the eta function to be expressed as a coupled system of real and imaginary components, each defined over partitioned summation blocks. Using this decomposition, a real—imaginary interaction structure is introduced, where intersection-type conditions between the real and imaginary parts are studied through a determinant-based formulation. The resulting system suggests a structured relationship between block interactions and phase behavior in the complex plane. The framework provides a new perspective on the analytic structure of η(s)through interval scaling, phase coupling, and approximate geometric recursion. In particular, the model highlights a special role of the parameter σ=1/2within the interaction structure, emerging from symmetry considerations in the decomposed representation. This work is exploratory in nature and aims to develop a structured analytical model for studying oscillatory behavior in alternating Dirichlet series via block decomposition techniques.
Category: Functions and Analysis

[2] viXra:2605.0059 [pdf] submitted on 2026-05-15 21:48:50

Finite Product Limits for Elementary and Gamma Functions

Authors: Edigles Guedes
Comments: 22 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)

This paper presents a series of novel finite product limit representations for several fundamental functions: power function, gamma function and reciprocal gamma function.
Category: Functions and Analysis

[1] viXra:2605.0018 [pdf] submitted on 2026-05-07 19:55:34

Asymptotic Regularization of Poles and Closed-Form Generalization for Arithmetic Progressions in the Gamma Function

Authors: Julinho Jorge Luís
Comments: 25 Pages.

This paper proposes a novel analytical framework for the Gamma and Factorial functions, extending their consistent application beyond the domain of positive real numbers. Traditionally, these functions encounter singularities (poles) at non-positive integers, limiting their continuity. Through an asymptotic regularization method, this work demonstrates that the ratio of Gamma functions can yield finite and unique values at these critical points, effectively bypassing traditional meromorphic constraints.The core contribution is the derivation of a universal closed-form formula for the product of arithmetic progressions, valid across the entire real line without the need for manual domain adjustments. Furthermore, the concept of a "Rising Gamma Function" ($check{Gamma}$) is introduced as a dual operator. By establishing the zero point as an inversion axis, a functional symmetry is revealed, integrating the properties of the function into a complete and continuous structure. This approach provides new insights into analytical continuity and simplifies calculations in complex analysis and number theory.Keywords: Gamma Function, Factorial, Asymptotic Regularization, Arithmetic Progressions, Functional Symmetry, Analytical Continuity.
Category: Functions and Analysis