/sci/ - Science & Math

https://arxiv.org/pdf/1911.10934.pdf
>An Explicit Identity of Sums of Powers of Complex Functions to Prove Riemann Hypothesis
>Dagnachew Jenber Negash
>(Submitted on 22 Nov 2019)

>This paper presents an absolutely explicit identity for solving sums of powers of complex functions with out one sum depends on the others. Via this sums of powers of complex functions, this paper proves Riemann Hypothesis.

https://arxiv.org/pdf/1911.02400.pdf
>Proof of the Collatz Conjecture
>Agelos Kratimenos
>(Submitted on 4 Nov 2019)

>Collatz Conjecture is one of the most famous, for its simple form, proposed more than eighty years ago. This paper presents a full attempt to prove the affirmative answer to the question proposed by the conjecture. In the first section, we propose a number of definitions utilized later on the proof. In the second section, we discover the formula for a characteristic function. This formula describes the functionality of the paths taken for each number based on the Collatz Sequence. In the last section, we prove that every number will eventually reach 1, using the characteristic function.

https://arxiv.org/pdf/1910.02954
>Prime numbers and the Riemann hypothesis
>Tatenda Kubalalika
>(Submitted on 7 Oct 2019)

>By considering the prime zeta function, we demonstrate in this note that the Riemann zeta function zeta(s) does not vanish for Re(s)>1/2, which proves the Riemann hypothesis.

https://arxiv.org/pdf/1909.10313
>A Proof of Riemann Hypothesis
>Tao Liu, Juhao Wu
>(Submitted on 19 Sep 2019)

>The ratio of the Riemann-Zeta function [math]W(s)=ζ(s)/ζ(1-s)[/math] maps the line of [math]s=1/2+it[/math] onto the unit circle in W-space. [math]|W(s)|=0[/math] gives the trivial zeroes of the Riemann-Zeta function [math]ζ(s)[/math]. In the range: [math]0<|W(s)|\neq1[/math], [math]ζ(s)[/math] does not have nontrivial zeroes. [math]|W(s)|=1[/math] is the necessary condition of the non-trivial zeros of the Riemann-Zeta function. Writing [math]s=σ+it[/math], in the range: [math]0\leqσ\leq1[/math], but [math]σ\neq 1/2[/math], even if [math]|W(s)|=1[/math], the Riemann-Zeta function [math]ζ(s)[/math] is non-zero. Based on these arguments, the non-trivial zeros of the Riemann-Zeta function [math]ζ(s)[/math] can only be on the [math]s=1/2+it[/math] critical line; therefore the proof of the Riemann Hypothesis.

https://arxiv.org/pdf/1909.07975.pdf
>A Proof of the Twin Prime Conjecture
>Berndt Gensel
>(Submitted on 17 Sep 2019)

>The twin prime conjecture 'There are infinitely many twin primes' is a very old unsolved mathematical problem. This paper develops a sieve to extract all twin primes on the level of their generators up to any limit. The sieve uses only elementary methodes. With this sieve the twin prime conjecture finally can be proved indirectly.

https://arxiv.org/pdf/1909.02205.pdf
>The answer to a conjecture on the twin prime
>Mbakiso Fix Mothebe, Dang Vo Phuc
>(Submitted on 5 Sep 2019)

>Let p_1=2,p_2=3,p_3=5,…,p_n,… be the ordered sequence of consecutive prime numbers in ascending order. For a positive integer m, denote by π(m) the number of prime numbers less than or equal to m. Let [] denote the floor or greatest integer function. In this paper, we show that for all n≥1:
>[math] \left[\frac{p^2_{n+1}}{n+1} \right]
\leq \pi\left(p^2_{n+1} \right). [/math]
>As a consequence, we see that there are infinitely many primes (Euclid's theorem). Then, we prove that if we let π_2(m), denote the number of twin primes not exceeding m, then for all n≥2:
>[math]\left[\frac{p^2_{n+3}}{3(n+2)} \right] \leq \pi_2\left(p^2_{n+3}
\right)[/math]
>and thereby prove the twin prime conjecture, namely that there are infinitely many prime numbers p for which p+2 is also a prime.