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[17] viXra:2202.0169 [pdf] replaced on 2024-03-09 23:46:13
Authors: James Edwin Rock
Comments: 1 Page.
The Mӧbius function µ(j)= -1,0,1 depending on whether j has an odd number of factors, a square factor, or an even number of factors. The Mertens function m(n) is j = 1 to n, ∑ µ(j). For all n, |m(n)| < (2)(sqrt(n)). |m(n)| = O(sqrt(n), and therefore the Riemann Hypothesis is true.
Category: Number Theory
[16] viXra:2202.0166 [pdf] submitted on 2022-02-25 19:16:08
Authors: Tae Beom Lee
Comments: 26 Pages. The revised version of the previous one, 'Two Proofs of Riemann Hypothesis by Vector Properties of Riemann Zeta Function'
The Riemann Hypothesis (RH) states that the non-trivial zeros of the Riemann Zeta Function (RZF) ζ(s) or the Dirichlet Eta Function(DEF) η(s) for a complex variable s = x + iy is of the form s = 0.5 + iy. In this thesis, we treat each term of the RZF as a vector. We showed some vector properties of the RZF by tracing term vectors. If there exist zeros whose real part is not 0.5, such as ζ(α + iβ) = ζ(1-α + iβ) = 0, the trajectory of ζ(α + iy) and ζ(1-α + iy) must intersect at the origin when y = β. To check if this can happen, we introduced the rubber strip model, and by using the Cauchy-Riemann differential equations, we induced a contradiction, ζ(s) = constant, which proves the RH. In appendices, we provided the source programs for visualizing vector traces of the RZF. We also suggested three other possible proofs of the RH for further studies.
Category: Number Theory
[15] viXra:2202.0158 [pdf] submitted on 2022-02-24 11:22:24
Authors: Arthur V. Shevenyonov
Comments: 2 Pages. a most minimalist, elementary approach
The oft-ventured yet elusive Riemann Hypothesis allows for some unparalleled, perfecting simplicity which arguably denies any more-economical means. An overlap obtains with prior work from a drastically different angle.
Category: Number Theory
[14] viXra:2202.0147 [pdf] replaced on 2022-05-31 20:33:22
Authors: Wing K. Yu
Comments: 5 Pages.
In this paper we will use a different way to prove that there exists at least a prime number p in between 2n and 3n where n is a positive integer. The proof extends the Bertrand’s postulate - Chebyshev’s theorem which states that a prime number exists between n and 2n. The method to prove this proposition is to analyze the binomial coefficient, a similar method used by Erdős in the proof of Bertrand’s postulate.
Category: Number Theory
[13] viXra:2202.0145 [pdf] submitted on 2022-02-22 20:50:06
Authors: Giovanni Di Savino
Comments: 4 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]
Thanks to an intuition and a strategy, Gauss showed that the same odd number, 101, is the sum of the even numbers with the odd numbers contained in the number 100. The strategy invented by Gauss is indicative to satisfy Goldbach's conjecture.
Category: Number Theory
[12] viXra:2202.0129 [pdf] submitted on 2022-02-20 10:30:01
Authors: Marko V. Jankovic
Comments: 30 Pages.
In this paper proof of the twin prime conjecture is going to be presented. Originally very difficult problem (in
observational space) has been transformed into a simpler one (in generative space) that can be solved. It will be shown
that twin primes could be obtained through two stage sieve process, and that will be used to show that exist infinite
number of twin primes. The same approach is used to prove Polignac's conjecture for cousin primes.
Category: Number Theory
[11] viXra:2202.0102 [pdf] submitted on 2022-02-14 23:15:45
Authors: Elizabeth Zhou, Ming Zhou
Comments: 3 Pages.
Here two novel expressions of the Dirichlet Beta function at 10 are provided.
Category: Number Theory
[10] viXra:2202.0087 [pdf] submitted on 2022-02-13 12:43:09
Authors: Arthur V. Shevenyonov
Comments: 4 Pages. a scheme that seems to be bridging & straddling
The Collatz conjecture, elusive as it may prove, has shown to be collated to other arcane results, notably ABC, Goldbach’s, and their generalized ilk. The present paper demonstrates that, whilst a straightforward proof scheme could (ironically) be a near trivial enterprise, the philosophical implications would posit more of a far-reaching agenda.
Category: Number Theory
[9] viXra:2202.0074 [pdf] submitted on 2022-02-12 10:59:59
Authors: Yuji Masuda
Comments: 1 Page.
The purpose of this short paper is to deepen the basic knowledge of prime numbers, focusing on the prime number generation formula. The prime number generating equation has been analyzed to some extent, and I would like to contribute to it.
Category: Number Theory
[8] viXra:2202.0071 [pdf] submitted on 2022-02-12 14:17:56
Authors: José Alcauza
Comments: 3 Pages.
In this paper, we find a curious and simple possible solution to the critical line of nontrivial zeros of Riemann zeta function.
Category: Number Theory
[7] viXra:2202.0064 [pdf] submitted on 2022-02-11 18:31:58
Authors: Vedant Lohia
Comments: 17 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]
The late Paul Erdős famously said that regarding the Collatz Conjecture, "Mathematics is not yet ready for such problems" and his words ring true to this day. This paper outlines the attempts made to prove or disprove the conjecture and refrains from restricting itself to a very mathematical audience. While surface level knowledge of such conjectures and their nature might satiate some, it is for those who wish to know more before a sea of notation and concepts overwhelms them that I dedicate this paper.
Category: Number Theory
[6] viXra:2202.0047 [pdf] submitted on 2022-02-09 19:24:07
Authors: Arthur V. Shevenyonov
Comments: 3 Pages. Prime-generating structures & symmetries
Dial symmetry, Complenarity, and #-scoring could prove productive as well as efficient means of [re]constructing primes. Seen as alternate constructs accommodating the same prime values, or otherwise pairing those implicitly involved, as p’=p+2k*9 or p’=2n+(2k+1)*9 (both capturing the laxer subset of p=2m+1), complenary values tend to follow the parity: #p’=#p=#(2m). Moreover, the early primes sequence appears to fit into Fibonacci-like regularities along much the same #-scoring lines.
Category: Number Theory
[5] viXra:2202.0037 [pdf] submitted on 2022-02-05 23:13:17
Authors: Andrea Prunotto
Comments: 5 Pages.
In this work, we introduce the concept of Fermat’s Urn, an urn containing three types of marbles, and such that it holds a peculiar constraint therein: The probability to get at least one marble of a given type (while performing multiple independent drawings) is equal to the probability not to get any marble of another type. Further, we discuss a list of implicit hypotheses related to Fermat's Equation, which would allow us to interpret this equation exactly as the mentioned constraint in Fermat's Urn. Then, we study the properties of this constraint in relation with the capability to distinguish the types of marbles within the urn, namely in case of the event ''to get at least one marble of each type''. Eventually, on the basis of a simple theorem related to this event, we prove that Fermat's Equation and Fermat's Urn may share those properties only if we perform at most two drawings from the urn. This result reflects then in the solution of Fermat's Equation.
Category: Number Theory
[4] viXra:2202.0026 [pdf] replaced on 2023-02-20 04:59:21
Authors: Ronald Danilo Chávez
Comments: 37 Pages.
This paper shows a very elementary way of counting the number of primes under a given number with total accuracy. Is the function π(x) if 25 ≤ x ≤ 1572.
Category: Number Theory
[3] viXra:2202.0021 [pdf] submitted on 2022-02-04 21:27:24
Authors: Jorma Jormakka
Comments: 24 Pages.
Zeros and the pole of the Riemann zeta function ζ(s) correspond to simple poles of the logarithmic derivative f(s) = d/ds ln ζ(s). In Re{s} > 1 the function f (s) has an absolutely convergent sum expression and an analytic continuation to the complex plane except for a discrete set of simple poles in the area Re{s} ≤ 1. Close to a pole sk the function f (s) is rk /(s−sk)+finite terms. Omitting the finite terms, we can evaluate this function into a Taylor series at the x-axis point x > 1. The absolute values of the coefficients of the Taylor series of each pole decrease as x−i for some i > 0 as a function of x. The absolute values of the coefficients of the Taylor series of f (s) decrease as a negative exponent of x when x grows. That means that all terms aix−i, ai ∈ IR, are cancelled by other terms in f (s) when x → ∞. These other terms must contain terms −aix−i. Such terms arise only from poles. It follows that in the sum of all poles of f (s), at the point x, poles must cancel other poles when x → ∞. The poles of f (s) in Re{s} ≥ 1 and Re{s} ≤ 0 are known. They are the only poles that give a negative coefficients of x−j , j > 0, while the remaining poles, the non-trivial zeros of ζ(s), give positive coefficients. It is shown that the poles of f (s) cancel when x → ∞ if and only if every pole sk at 0 < Re{sk } < 1 satisfies Re{s} = 1 2 , i.e., the Riemann Hypothesis is true.
Category: Number Theory
[2] viXra:2202.0007 [pdf] submitted on 2022-02-02 17:56:56
Authors: Julian Beauchamp
Comments: 1 Page. [Note by viXra Admin: Author's name must appear on the submitted article or the submission will be rejected without notice]
In this short paper I demonstrate a simple connection between primitive Pythagorean triples of the form {X, Y, Z=Y+1} and the squares of the Pell Numbers. I conjecture that when X is equal to one of the numerators of continued fraction convergents to sqrt 2, then and only then can Y or Z be a square, and only then a square Pell Number.
Category: Number Theory
[1] viXra:2202.0003 [pdf] submitted on 2022-02-01 08:04:08
Authors: V. Barbera
Comments: 2 Pages.
This paper presents an exact elementary formula for the Goldbach function.
Category: Number Theory
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