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| viXra.org > Functions and Analysis > 2411 |
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[5] viXra:2411.0142 [pdf] submitted on 2024-11-22 21:38:51
Authors: Mostafa Senhaji
Comments: 59 Pages.
The Riemann zeta function, symbol of the convergence between complex analysis and number theory, occupies a central place in modern mathematics. Since its introduction by Bernhard Riemann in 1859, this function has established itself as an essential tool for understanding the distribution of prime numbers. This work is distinguished by the ambition to propose a formal proof of the Riemann Hypothesis (HR), a conjecture which intimately links the non-trivial zeros of ��(��) to the structure of integers. By exploring its analytical, geometric and arithmetic aspects, this text constitutes a significant step forward in the quest for this proof. The objective of this work is twofold: on the one hand, to offer an in-depth presentation of the fundamental properties of ��(��), such as its functional equation, its Euler product, and the symmetry of its zeros; on the other hand, rigorously demonstrate that all non-trivial zeros of ��(��) lie on the critical line ℜ(��)=1/2. In this way, it is aimed not only at specialists but also at mathematics enthusiasts, by offering them a unique and detailed perspective on one of the greatest mathematical mysteries. Through a rigorous methodology, illustrated by demonstrations by contradiction and analytical visualizations, this text highlights the profound consequences of HR, not only on number theory, but also on other branches of mathematics. It invites the reader to explore this hidden harmony that links the analytical properties of complex functions and the regularity of prime numbers.
Category: Functions and Analysis
[4] viXra:2411.0063 [pdf] submitted on 2024-11-08 18:31:49
Authors: Edgar Valdebenito
Comments: 2 Pages.
We estimate the Laplace constant using Lambert's W function.
Category: Functions and Analysis
[3] viXra:2411.0022 [pdf] submitted on 2024-11-03 23:00:55
Authors: Theophilus Agama
Comments: 5 Pages.
In this note, we introduce and develop the analysis of the fractional invariance. This analysis is used for estimating the partial sums of arithmetic functions $f:mathbb{N}longrightarrow mathbb{R}$ of the form $sum limits_{substack{nleq xin mathbb{A}}}f(n)$ for $mathbb{A}subseteq mathbb{N}$. This analysis can be applied to a broad class of arithmetic functions.
Category: Functions and Analysis
[2] viXra:2411.0016 [pdf] submitted on 2024-11-03 22:49:27
Authors: Edigles Guedes
Comments: 3 Pages.
In this paper, we present an integral representation involving trigonometric functions and variable transformation techniques to turn it into a triple integral. The proposed integral is initially simplified using trigonometric identities, so we rewrite the original integral in terms of a three-variable integral representation. The main theorem demonstrates the equivalence between the initial integral and the resulting triple integral, illustrating the applicability of trigonometric identities in calculations of complicated integrals.
Category: Functions and Analysis
[1] viXra:2411.0002 [pdf] submitted on 2024-11-01 16:05:44
Authors: Juan Elias Millas Vera
Comments: 1 Page.
I taught myself Calculus seriously the last months and I want to share some thoughts about Real constants Calculus with a single variable. It will may be mostly know this results by professionals but, in any case it could be interesting to share my conclusions in a short paper.
Category: Functions and Analysis
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