A Modification to the Novel Toy Model of the Riemann Zeta Function Roots Equation

Lam, Kai Shun (2024) A Modification to the Novel Toy Model of the Riemann Zeta Function Roots Equation. International Journal of Mathematics and Statistics Studies, 12 (4). pp. 16-35. ISSN 2053-2229 (Print), 2053-2210 (Online)

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Abstract

It is true that my previous series of papers about the Riemann Hypothesis has raised lots of discussions or the concerns among the mathematics society. Indeed, some professionals may think that the Riemann Zeta function at s = 1 or ∑_(n=1)^∞▒1/n is actually divergent and may tend to an infinity, how it can be acted as a denominator in a proof or something like 1/∞ ? This author’s idea is there may be no need to evaluate such kind of summation as one may consider the sum just as an (imaginary) constant (but NOT the complex number). Or we may only consider such summation as a kind of improper integrals with a black-boxed answer. In other words, the true final value of the summation ∑_(n=1)^∞▒1/n is NOT an concerning serious issue. The function ∑_(n=1)^∞▒1/n^s is in practice a smooth function and equals to log N or the logarithmic function for s = 1 or have the infinite differentiable properties except n = 0, that is why the commercial software Maple Soft can compute the Zeta function’s associated Taylor series successfully. In fact, from the above series together with the harmonic properties, one may calculate the artifical Zeta function’s root model equation such as the 0.5 +/- β*cot(ln(x))/(αx+1)n (will be discussed in my next paper of the series). Indeed, this is NOT the best model. Actually, the true Riemann Zeta non-trivial Root model equation is: {z = x + yI | z∈0.5±(y_1⩽y⩽y_2 )I} where y1= (±cot(k))/ln(k) , y2 = (±tan(k))/ln(k) with 0<k<2π. Certainly, this author cannot avoid mistakes. Some of the defects in the previous paper such as one in the linguistic proof of the RH will be stated and amended. Lastly, this author wants to remind that the present series of paper is first to establish a model equation for zeta function’s roots and the Plank’s like constant. This author then applies the telescopic and logarithmic methods for a 3 cases proof of the RH. Also, we have a Matlab code for the evaluation of RH complex contour integral. Finally, the RH problem will be transformed into a linguistic one and hence we may have solved the Riemann Hypothesis issue completely through a 4 cases of truth table collaborations with the logical inference [7] & [15] to the causality (structural) organization of the sentences [16] or just named as “Logical & Organized Context”.

Item Type:	Article
Subjects:	Q Science > QA Mathematics
Depositing User:	Professor Mark T. Owen
Date Deposited:	29 May 2024 11:08
Last Modified:	29 May 2024 11:08
URI:	https://tudr.org/id/eprint/3043

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