arXiv subjects

Nikos Mantzakouras

Publications and source records attributed to Nikos Mantzakouras.

7 recordsLinked to original sources

arXiv2025

Conditions for solving polynomial equations using algebraic and hypergeometric functions

In this paper, we focus on clarifying the concept of solving equations of degree greater than six using continuous functions or hypergeometric functions and providing another proof of the non-existence of algebraic solutions for equations of degree greater than four. According to the Kolmogorov-Arnold theorem, we will prove that equations of degree greater than five cannot be solved without special conditions between their coefficients using hypergeometric functions. However, we prove that trinomial equations of general form can in general be solved using hypergeometric functions.

math.GM↗

arXiv2025

Holder continuity of an alternating Erdos series on prime K-tuples

This open problem, first posed by Erdοs, was further explored by Terence Tao. Tao work shows that the series can converge conditionally, but only under a sufficiently strong form of the Hardy-Littlewood conjecture for k-primary pairs. Based on this, we offer a new method leading to a representation of the series as a Riemann-Stieltjes integral or a tightly coupled prime counting function. We rigorously analyze this integral by decomposing it into principal and error terms, applying integration by parts in the Stieltjes sense, and defining the error terms. Assuming the Riemann hypothesis, we investigate the Hοlder continuation of ψ(x) in the asymptotic form ψ(x) = x+O(x 1/2), and introduce a test function g(x) = e^( iπx) e^( -λx) , which is smooth and Lipschitz. Applying Young's criterion, we show that the integral converges. Moreover , we prove that the integral converges perfectly for λ > 3 2 , based on sharp bounds on the error terms. Our results are supported by fractional Sobolev integrations and justify the use of Young's inequality under generalized Holder conditions.

math.GM↗

arXiv2025

Diophantine FLINT-HILLS series

We prove the convergence of the FLINT-HILLS series and establish new criteria for a similar type of diophantine or lacunary series, which faces issues due to spaced long terms coming from the trigonometric nature of functions, e.g., cosecant in the FLINT-HILLS series. We connect the FLINT-HILLS series to the Fermi-Dirac integral via the Riemann-Stieltjes integral and YOUNG'S inequality criteria but also proved that the upper bound of the irrationality measure of pi is equal or lower than 2.5 expected if the FLINT-HILLS series converged.

math.GM↗

arXiv2023

Hypothesis of Riemann is rejected by definition

Hypothesis of Riemann is rejected by definition, because ζ(s), where s zeros of ζ(s)=0, is not be equal by definition to the particular sum, which it assumes to be equal. R(s) = 1/2 holds only for the zeros of ζ(s) = 0 and for the zeros of certain related functions. However, it does not hold for certain special generalized functions of ζ(), such the Zeta Hurwitz functions and their sums.

math.GM↗

arXiv2022

Generalized solutions of Polynomial and Transcendental Equations by the strong method of (Generalized Iterative Method approximation of Roots (GRIM) )

In this paper, we explain a new Iterative Method-Fixed Point and develop its convergence theory for finding approximate solutions of nonlinear equations in the setting of Banach spaces. First, we discuss the convergence analysis of our method by separating the equation into functions from which it is derived and the remaining part of the equation, according to the Generalized Theorem[1] for solving polynomial and transcendental equations. Solving an equation, either polynomial or transcendental, is solved only by categorizing the roots and not by some procedure of searching in individual intervals and this way was followed in the past by all existing methods. But the methods developed in the past are limited to isolated intervals without general acceptance. Finally, the method is strengthened by Newton's method to speed up the finding of the roots. At the end of the analysis, we give several examples of the application of the method.

math.GM↗

arXiv2021

New Formulas for the Euler-Mascheroni Constant and other Consequences derived from the Acceptance of Hyperbolicity of Jensen Polynomials and the Analysis of the Tur\'an Moments for the {\xi}-Function

The Euler-Mascheroni constant is calculated by three novel representations over these sets respectively: 1) Tur\'an moments, 2) coefficients of Jensen polynomials for the Taylor series of the Riemann Xi function at s=1/2+i.t and 3) even coefficients of the Riemann Xi function around s=1/2. These findings support the acceptance of the property of hyperbolicity of Jensen polynomials within the scope of the Riemann Hypothesis due to exactness on the approximations calculated not only for the Euler-Mascheroni constant but also for the Bernoulli numbers and the even derivatives of the Riemann Xi function at s=1/2. The new formulas are linked to similar patterns observed in the formulation of the Akiyama-Tanigawa algorithm based on the Gregory coefficients of second order and lead to understanding the Riemann zeta function as a bridge between the Gregory coefficients and other relevant sets

math.GM↗

arXiv2021

Solve Polynomial and transcendental Equations with use Generalized Theorem (Method Lagrange)

The great innovation of the Generalized Theorem is that it gives us the philosophy to work out the knowledge that the number of roots of an equation depends on the subfields of the functional terms of the equation they generate. Thus, the final field of the roots of the equation will be the union of these subfields. We have a wide range of applications by solving hyperbolic in all sciences, especially in physics and chemistry. In this paper, we solve the generalized trinomial and some important applications in physics and astronomy like the generalized solution of the Kepler equation.

math.GM↗