A proof of the Riemann hypothesis based on the Koch theorem, on primes in short intervals, and distribution of nontrivial zeros of the Riemann zeta function

Part One: Let define the truncation of the logarithmic integral $Li(x)$ as $$\pi^{*}(x,M)=\frac{x}{\log x}\sum_{n=0}^{M}\frac{n!}{\log^{n}x}.$$ First, we prove $\pi^{*}(x,M)\leq Li(x)<\pi^{*}(x,M+1)$ which implies that the point of the truncation depends on x, Next, let $\alpha_{L,M}=x_{M+1}/x_{M}$. We prove that $\alpha_{L,M}$ is greater than $e$ for $M<\infty$ and tends to $\alpha_{L,\infty}=e$ as $M \to \infty$. Thirdly, we prove $$M=\log x-2+O(1)\texttt{ for }x\geq24.$$ Finally, we prove $$Li(x)-\pi^{*}(x,M)<\sqrt{x}\texttt{ for }x\geq24.$$ Part Two: Let define $$\pi^{*}(x,N)=\frac{x}{\log x}\sum_{n=0}^{N}\frac{n!}{\log^{n}x}$$ where we proved that the pair of numbers $x$ and $N$ in $\pi^{*}(x,N)$ satisfy inequalities $\pi^{*}(x,N)<\pi(x)<\pi^{*}(x,N+1)$, and the number $N$ is approximately a step function of the variable $\log x$ with a finite amount of deviation, and proportional to $\log x$. We obtain much more accurate estimation $\pi(x)-\pi^{*}(x,N)$ of prime numbers, the error range of which is less than $\sqrt{x}$ for $x\geq10^{3}$ or less than $x^{1/2-0.0327283}$ for $x\geq10^{41}$. Part Three: We show the closeness of $Li(x)$ and $\pi(x)$ and give the difference $|\pi(x)-Li(x)|$ being less than or equal to $c\sqrt{x}\log x$ where $c$ is a constant. Further more, we prove the estimation $Li(x)=\pi^{*}(x,N)+O(\sqrt{x})$. Hence we obtain $\pi(x)=Li(x)+O(\sqrt{x})$ so that the Riemann hypothesis is true. Part Four: Different from former researches on the distribution of primes in short intervals, we prove a theorem: Let $\Phi(x)=\beta x^{1/2}$, $\beta>0$, and $x\geq x_{\beta}$ which satisfies $(\log x_{\beta})^{2}/x_{\beta}^{0.0327283}\leq\beta$. Then there are $$\frac{\pi(x+\Phi(x))-\pi(x)}{\Phi(x)/\log x}=1+O(\frac{1}{\log x})$$ and $$\lim_{x \to \infty}\frac{\pi(x+\Phi(x))-\pi(x)}{\Phi(x)/\log x}=1.$$Comment: 95 page

Advancing Learner Autonomy in Tefl Via Collaborative Learning

Learner autonomy has been defined as \u27a capacity to control important aspects of one\u27s learning\u27(Benson, 2013, p. 852). In the teaching of additional languages, learner autonomy dates back at least to the 1970s. For instance, Trim, who was a leader in the teaching of additional languages in Europe, stated that a goal of language education was to: make the process of language learning more democratic by providing the con- ceptual tools for the planning, construction and conduct of courses closely geared to the needs, motivations and characteristics of the learner and enabling him [sic] so far as possible to steer and control his own progress. (1978, p. 1

On the representation of even numbers as the sum and difference of two primes and the representation of odd numbers as the sum of an odd prime and an even semiprime and the distribution of primes in short intervals

The representation of even numbers as the sum of two primes and the distribution of primes in short intervals were investigated and a main theorem was given out and proved, which states: For every number $n$ greater than a positive number $n_{0}$, let $q$ be an odd prime number smaller than $\sqrt{2n}$ and $d=2n-q$, then there is always at least an odd number $d$ which does not contain any prime factor smaller than $\sqrt{2n}$ and must be an odd prime number greater than $2n-\sqrt{2n}$. Then it was proved that for every number $n$ greater than 1, there are always at least a pair of primes $p$ and $q$ which are symmetrical about the number $n$ so that even numbers greater than 2 can be expressed as the sum of two primes. Hence, the Goldbach's conjecture was proved. Also theorems of the distribution of primes in short intervals were given out and proved. By these theorems, the Legendre's conjecture, the Oppermann's conjecture, the Hanssner's conjecture, the Brocard's conjecture, the Andrica's conjecture, the Sierpinski's conjecture and the Sierpinski's conjecture of triangular numbers were proved and the Mills' constant can be determined. The representation of odd numbers as the sum of an odd prime number and an even semiprime was investigated and a main theorem was given out and proved, which states: For every number $n$ greater than a positive number $n_{0}$, let $q$ be an odd prime number smaller than $\sqrt{2n}$ and $d=2n+1-2q$, then there is always at least an odd number $d$ which does not contain any odd prime factor smaller than $\sqrt{2n}$ and must be a prime number greater than $2n+1-2\sqrt{2n}$. Then it was proved that for every number $n$ greater than 2, there are always at least a pair of primes $p$ and $q$ so that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime. Hence, the Lemoine's conjecture was proved.Comment: 265 page

An autonomous agent for learning spatiotemporal models of human daily activities

Activities of Daily Living (ADLs) refer to activities performed by individuals on a daily basis. As ADLs are indicatives of a person's habits, lifestyle, and well being, learning the knowledge of people's ADL routine has great values in the healthcare and consumer domains. In this paper, we propose an autonomous agent, named Agent for Spatia-Temporal Activity Pattern Modeling (ASTAPM), being able to learn spatial and temporal patterns of human ADLs. ASTAPM utilises a self-organizing neural network model named Spatiotemporal - Adaptive Resonance Theory (ST-ART). ST-ART is capable of integrating multimodal contextual information, involving the time and space, wherein the ADL are performed. Empirical experiments have been conducted to assess the performance of ASTAPM in terms of accuracy and generalization.NRF (Natl Research Foundation, S’pore)Published versio