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[8] viXra:2608.0042 [pdf] submitted on 2026-08-11 01:45:32
Authors: Abdelmajid Ben Hadj Salem
Comments: 10 Pages. Submitted to Ramanujan Journal. Comments welcome.
In 1859, Georg Friedrich Bernhard Riemann had announced the following conjecture, called the Riemann hypothesis : textit{The nontrivial roots (zeros) $s=sigma+it$ of the zeta function, defined by:} $$zeta(s) = sum_{n=1}^{+infty}frac{1}{n^s},,mbox{for}quad Re(s)>1$$ textit{have real part} $sigma= frac{1}{2}$. In this paper, I give the proof that $sigma= frac{1}{2}$ using an equivalent statement of the Riemann hypothesis: the Dirichlet $eta$ function and the functional relation verified by the zeta function.
Category: Number Theory
[7] viXra:2608.0040 [pdf] submitted on 2026-08-09 21:29:01
Authors: Viktor Voevodov
Comments: 8 Pages. (Note by viXra Admin: Please cite and list scientific references and submit article written with AI assistance to ai.viXra.org)
This paper proposes a fundamentally new approach to investigating the structure of the set of natural numbers based on the concept of alternative and connected numerical sets, one of which is the set of prime numbers. The method of dynamic sieves is described. By utilizing the proposed kinematic model and the theory of connected sets, a theorem is proven on the infinity of sets of n-tuple primes of arbitrary length and configuration. While the theory of connected sets expands the theorem, the theory of alternative sets deepens it, rendering the theorem global.
Category: Number Theory
[6] viXra:2608.0038 [pdf] submitted on 2026-08-09 13:18:50
Authors: George Athanasiou
Comments: 19 Pages.
We develop a residue-based method for recovering the coefficients of aDirichlet series from the singularities of its analytic continuation. Startingfrom the classical vertical-line inversion formula, we introduce a conformaltransformation that converts coefficient extraction into a contour integraland yields a spectral representation in terms of poles and residues. Weapply this framework to the von Mangoldt, prime characteristic, Möbius,and Euler totient functions, obtaining representations involving the zerosand poles of the Riemann zeta function. We also introduce generalized vonMangoldt functions associated with arbitrary Dirichlet series and extendthe approach to finite and Dedekind zeta functions, leading to formulas forpoint counts, prime-ideal counting functions, and characteristic functionsover number fields. Finally, we examine prime-pair, Goldbach, andprime-tuple counting functions and derive conditional error estimatesunder the Riemann hypothesis. These results present contour integrationand zeta-function factorizations as a unified mechanism for translatinganalytic spectral data into arithmetic information.
Category: Number Theory
[5] viXra:2608.0029 [pdf] submitted on 2026-08-08 02:41:13
Authors: Theophilus Agama
Comments: 10 Pages.
An addition chain of length $h$ that leads to an integer $ngeq 2$ is a strictly increasing sequence of positive integers $s_0=1,s_1=2,ldots,s_h=n$ such that $s_i=s_j+s_k$ with $i>jgeq kgeq 0$ for each $iin {1,ldots,h}$. We denote the minimal length of an addition chain that leads to an integer target $cdot$ by $ell(cdot)$. In this paper, we introduce the concept of emph{quasi closed} addition chains and show that numbers $ngeq 2$ that admit a minimal-length quasi closed chains that are also a minimal-length addition (quasi complete numbers) satisfy the Scholz conjecture, precisely the inequality $$ ell(2^n-1)leq n-1+ell(n).$$
Category: Number Theory
[4] viXra:2608.0025 [pdf] submitted on 2026-08-08 02:21:38
Authors: Fischel Bloom
Comments: 2 Pages. (Note by viXra Admin: Please cite and listed scientific references)
We solve the open problem of proving Fermat’s Last Theorem (FLT) for n > 2 using only tools available to Fermat. We infer from Euclid’s formula an n-th power generalization that is equivalent to equation z^n− y^n = x^n. This generalization clearly shows that FLT is correct for integral {(z,y,x)}with integral n>2.
Category: Number Theory
[3] viXra:2608.0011 [pdf] submitted on 2026-08-03 19:16:21
Authors: Igor Hrnčić
Comments: 3 Pages.
This paper disproves the Riemann Hypothesis by a different method.
Category: Number Theory
[2] viXra:2608.0007 [pdf] submitted on 2026-08-02 00:36:14
Authors: Taha Muhammad
Comments: 3 Pages.
For centuries, the existence of a Perfect Cuboid—an Euler Perfect Box where the three edges, three face diagonals, and the internal space diagonal are all positive integers—has remained one of the most resilient open challenges in number theory. This manuscript delivers a definitive proof establishing the absolute non-existence of such a system. By decomposing the structural framework into three exhaustive geometric classes (scaloid, isosceles, and equilateral), we implement an original polynomial transformation to isolate unbreakable irrationality barriers. We demonstrate that the arithmetic parameters governing the system are mutually exclusive across all integer domains, proving that an Euler Perfect Box is structurally impossible.
Category: Number Theory
[1] viXra:2608.0002 [pdf] submitted on 2026-08-02 00:16:55
Authors: Johannes Debouto
Comments: 13 Pages. In French
The field of real numbers is essential to many scientific disciplines. The first formal constructions—which appeared long after the field's initial discovery—involve properties crucial to its modern application, most notably the property of completeness. One of the most famous of these constructions implicitly relies on the Cantor-Dedekind axiom, a concept that has often sparked controversy within the mathematical community. This work aims to demonstrate the validity of the Riemann hypothesis within the axiomatic framework of ZF plus the Cantor-Dedekind axiom, utilizing a result concerning Euler's totient function derived from the study of cyclotomic polynomials.
Category: Number Theory
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