[14] **viXra:1909.0285 [pdf]**
*submitted on 2019-09-13 19:27:39*

**Authors:** Alfredo Olmos, R. Romyna Olmos

**Comments:** 7 Pages.

In this article we study some characteristics of polygonal numbers, which are the positive integers that can be ordered, to form a regular polygon.
The article is closed, showing the relation of the polygonal numbers, with the Beta function when expressing any polygonal number, as a sum of terms of the Beta function.

**Category:** Number Theory

[13] **viXra:1909.0276 [pdf]**
*submitted on 2019-09-14 04:59:40*

**Authors:** Toshiro Takami

**Comments:** 3 Pages.

I tried to find a new expression for zeta odd-numbers, but it seems to have remained a general expression.
However, it may be a new expression and will be published here.

**Category:** Number Theory

[12] **viXra:1909.0232 [pdf]**
*replaced on 2019-09-14 01:48:36*

**Authors:** Toshiro Takami

**Comments:** 21 Pages.

I wondered why I would do this and I traced the course. However, I did not understand much.
Numerically, they match perfectly.
Also, this is considered to indicate that ζ(5), ζ(7), ζ(9), ζ(11), ζ(13)........ζ(197), ζ(199) are irrational numbers. Because, ζ(3) is irrational number.
It can also be said that it is expressed by an expression using π2.

**Category:** Number Theory

[11] **viXra:1909.0231 [pdf]**
*replaced on 2019-09-12 04:45:05*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

ζ(4), ζ(6), ζ(8).......ζ(52), ζ(54) considered.
From these equations, it can be said that ζ(4), ζ(6), ζ(8).......ζ(52), ζ(54) are irrational numbers. ζ(56),ζ(58) etc. can also be expressed by these equations.
Because I use π2, these seem to be irrational numbers.
The fact that the even value of ζ(2n) is an irrational number can be said from the fact that
each even value of ζ(2n) is an irrational number in addition to π2.

**Category:** Number Theory

[10] **viXra:1909.0178 [pdf]**
*submitted on 2019-09-08 12:33:13*

**Authors:** Shekhar Suman

**Comments:** 5 Pages.

Analytic continuation by hadamard product is strictly monotonic which implies RH

**Category:** Number Theory

[9] **viXra:1909.0165 [pdf]**
*replaced on 2019-09-11 01:57:47*

**Authors:** Sitangsu Maitra

**Comments:** 4 pages

Proof of Goldbach's strong conjecture in a different way

**Category:** Number Theory

[8] **viXra:1909.0154 [pdf]**
*submitted on 2019-09-07 13:41:13*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is on primes3.

**Category:** Number Theory

[7] **viXra:1909.0103 [pdf]**
*submitted on 2019-09-05 18:48:43*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is on primes.

**Category:** Number Theory

[6] **viXra:1909.0059 [pdf]**
*submitted on 2019-09-03 23:11:41*

**Authors:** Ayal Sharon

**Comments:** 32 Pages. Approx. 7500 words, and approx. 130 references in the bibliography

Riemann's "analytic continuation" produces a second definition of the Zeta function, that Riemann claimed is convergent throughout half-plane $s \in \mathbb{C}$, $\text{Re}(s)\le1$, (except at $s=1$). This contradicts the original definition of the Zeta function (the Dirichlet series), which is proven divergent there. Moreover, a function cannot be both convergent and divergent at any domain value. In other mathematics conjectures and assumed-proven theorems, and in physics, the Riemann Zeta function (or the class of $L$-functions that generalizes it) is assumed to be true. Here the author shows that the two contradictory definitions of Zeta violate Aristotle's Laws of Identity, Non-Contradiction, and Excluded Middle. The of Non-Contradiction is an axiom of classical and intuitionistic logics, and an inherent axiom of Zermelo-Fraenkel set theory (which was designed to avoid paradoxes). If Riemann's definition of Zeta is true, then the Zeta function is a contradiction that causes deductive "explosion", and the foundation logic of mathematics must be replaced with one that is paradox-tolerant. If Riemann's Zeta is false, it renders unsound all theorems and conjectures that falsely assume that it is true. Riemann's Zeta function appears to be false, because its derivation uses the Hankel contour, which violates the preconditions of Cauchy's integral theorem.

**Category:** Number Theory

[5] **viXra:1909.0038 [pdf]**
*submitted on 2019-09-02 12:25:38*

**Authors:** Shekhar Suman

**Comments:** 5 Pages.

Modulus of Hadamard product is shown increasing which proves the Riemann Hypothesis

**Category:** Number Theory

[4] **viXra:1909.0027 [pdf]**
*submitted on 2019-09-01 12:06:47*

**Authors:** Francis Maleval

**Comments:** 1 Page.

The sieve of the addition of two prime numbers and the sieve of the product of two natural numbers are linked by a paradox of symmetrical objects. Goldbach's conjecture, additive version of a property of primes, would then have no chance being demonstrated if its multiplicative alter ego remained impenetrable to the disorder of prime numbers.

**Category:** Number Theory

[3] **viXra:1909.0019 [pdf]**
*submitted on 2019-09-01 21:24:11*

**Authors:** Zeolla Gabriel Martín

**Comments:** 4 Pages.

This document exposes the construction of infinite patterns for prime numbers smaller than P.
In this case, the pattern for prime numbers less than 11 is graphic.

**Category:** Number Theory

[2] **viXra:1909.0013 [pdf]**
*replaced on 2019-09-07 05:20:27*

**Authors:** Toshiro Takami

**Comments:** 7 Pages.

Up to now, I have tried to expand this equation and prove Riemann hypothesis with the equation of cos, sin, but the proof was impossible.
However, I realized that a simple formula before expansion can prove it.
The real value is 0 only when the real part of s is 1/2. Non-trivial zeros must always have a real value of zero.

**Category:** Number Theory

[1] **viXra:1909.0010 [pdf]**
*submitted on 2019-09-01 01:13:44*

**Authors:** Kurmet Sultan

**Comments:** 1 Page. This Russian version of the article.

The article reports on the new patterns of modular arithmetic.

**Category:** Number Theory