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

Connectivity-guaranteed and obstacle-adaptive deployment schemes for mobile sensor networks

Mobile sensors can relocate and self-deploy into a network. While focusing on the problems of coverage, existing deployment schemes largely over-simplify the conditions for network connectivity: they either assume that the communication range is large enough for sensors in geometric neighborhoods to obtain location information through local communication, or they assume a dense network that remains connected. In addition, an obstacle-free field or full knowledge of the field layout is often assumed. We present new schemes that are not governed by these assumptions, and thus adapt to a wider range of application scenarios. The schemes are designed to maximize sensing coverage and also guarantee connectivity for a network with arbitrary sensor communication/sensing ranges or node densities, at the cost of a small moving distance. The schemes do not need any knowledge of the field layout, which can be irregular and have obstacles/holes of arbitrary shape. Our first scheme is an enhanced form of the traditional virtual-force-based method, which we term the Connectivity-Preserved Virtual Force (CPVF) scheme. We show that the localized communication, which is the very reason for its simplicity, results in poor coverage in certain cases. We then describe a Floor-based scheme which overcomes the difficulties of CPVF and, as a result, significantly outperforms it and other state-of-the-art approaches. Throughout the paper our conclusions are corroborated by the results from extensive simulations

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

Heavy and light flavor jet quenching at RHIC and LHC energies

The Linear Boltzmann Transport (LBT) model coupled to hydrodynamical background is extended to include transport of both light partons and heavy quarks through the quark-gluon plasma (QGP) in high-energy heavy-ion collisions. The LBT model includes both elastic and inelastic medium-interaction of both primary jet shower partons and thermal recoil partons within perturbative QCD (pQCD). It is shown to simultaneously describe the experimental data on heavy and light flavor hadron suppression in high-energy heavy-ion collisions for different centralities at RHIC and LHC energies. More detailed investigations within the LBT model illustrate the importance of both initial parton spectra and the shapes of fragmentation functions on the difference between the nuclear modifications of light and heavy flavor hadrons. The dependence of the jet quenching parameter $\hat{q}$ on medium temperature and jet flavor is quantitatively extracted.Comment: 6 pages, 6 figure