Functions and Analysis

2408 Submissions

[6] viXra:2408.0127 [pdf] replaced on 2024-12-02 04:05:16

A Fourier Derivative Collocation Method for the Solution of the Navier—Stokes Problem

Authors: Daniel Thomas Hayes
Comments: 6 Pages.

A proposed solution to the millennium problem on the existence and smoothness of the Navier--Stokes equations.
Category: Functions and Analysis

[5] viXra:2408.0114 [pdf] submitted on 2024-08-27 20:13:18

Convergence of a Rather Particular Series

Authors: Mame Goumba Amar, Aba Lo Amar
Comments: 2 Pages.

This article aims to examine the convergence of a particular series. This series is defined by a sequence governed by a recurrence relation which we will analyze in detail. Its specificity lies in the fact that it is made up of blocks whose terms are selected according to a jump pattern, which accentuates its elegance. The establishment of the described form gives this series remarkable and distinctive properties. We will thus explore the convergence of this series and the interesting relationships which link the blocks together.
Category: Functions and Analysis

[4] viXra:2408.0104 [pdf] submitted on 2024-08-26 02:11:24

Connected Prime Digit Factorial Occurrences to The Riemann Zeta Function

Authors: Parker Emmerson
Comments: 8 Pages.

This paper explores the theoretical relationship between the frequency of prime digits in factorial representations and the non-trivial zeros of the Riemann Zeta function. By defining the prime digit frequency within fac- torials and aggregating these frequencies, we propose a hypothesis where such aggregated prime digit frequencies exhibit periodic patterns that mirror the distribution of the non-trivial zeros of the Riemann Zeta func- tion. Utilizing Fourier transform analysis, we identify periodic compo- nents in the digit frequencies that may correspond to these zeros. Sta- tistical tests, including Chi-Squared and Kolmogorov-Smirnov tests, are employed to validate this connection. This study suggests that the nature of prime digit frequencies in number sequences, such as factorials, may reflect deeper mathematical structures influenced by the Riemann Zeta functions zeros.
Category: Functions and Analysis

[3] viXra:2408.0100 [pdf] replaced on 2024-09-26 15:50:33

Extended Proof of the Collatz Conjecture with Quasi-Induction

Authors: Hans Rieder
Comments: 4 Pages.

In this manuscript, we present an extended proof of the Collatzconjecture, based on the novel approach of quasi-induction and a detailed analysis of the shrinking rate. The logarithmic approach plays a central role in demonstrating that the sequence continuously shrinks on average and eventually reaches the number 1. In addition, numerical computations on GitHub are mentioned to support these theoretical results.
Category: Functions and Analysis

[2] viXra:2408.0091 [pdf] submitted on 2024-08-22 15:57:18

The Nature of Semi-Exponentials

Authors: Warren D. Smith
Comments: 9 Pages.

A "semi-exponential" is a function F(z) such that F(F(z))=exp(z).We show that(a) no entire-analytic semi-exponential F(z) exists;(b) no semi-exponential F(z) exists that is analytic within any interior-connected domainthat includes both the real axis, and all complex Q obeying Q=exp(Q), in its interior, and whichmaps reals→reals;(c) Analytic semiexponentials do exist that map most reals to complex numbers and which have non-analytic points;(d) We also construct a useful piecewise-analytic real→real semi-exponentialsuch that F, F', and F'' all are continuous,and F(x) is strictly increasing and strictly concave-∪, for all real x;and indeed the domain of definition of this F(z) may beslightly expanded to a long and thin complex set that includes the real axis in its interior,albeit then F becomes discontinuous at an infinite set of nonreal points.(e) But we show that no piecewise-analytic, with piece boundaries being nonemptyrectifiable differentiable curves, semi-exponentialthat maps reals→reals can be defined within any domain that includes thestrip 0≤im(z)<π.Many of our arguments may be repurposed for many other "semi" functions besides the exponential.Finally (f) we show that a real-valuedC∞-smooth,strictly increasing, strictly concave-∪semi-exponential exists, which under certain asymptotic analyticity demands is unique.
Category: Functions and Analysis

[1] viXra:2408.0078 [pdf] submitted on 2024-08-18 21:56:24

The Proof of the Riemann Hypothesis

Authors: Zuher El Ahmed
Comments: 5 Pages.

The problem of finding the zeros of functions is one of the important issues in mathematics, and I would not be exaggerating if I said that all of mathematics is based on such problems. Here, curiosity struck me to understand the mechanism or the secret behind these functions. I never expected that this curiosity would lead me to encounter an important function like the Riemann zeta function, starting from the Taylor and Maclaurin series, which at least enabled me to find a function that links the point belonging to a certain domain and the values of this domain with an exponential function, as demonstrated in the proof. In conclusion, I believe that if this function cannot find the zeros of the Riemann zeta function, it will at least allow us to look at the zeta function from another perspective that is easier to deal with.
Category: Functions and Analysis