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

Effect of isoelectronic doping on honeycomb lattice iridate A_2IrO_3

We have investigated experimentally and theoretically the series (Na$_{1-x}$Li$_{x}$)$_{2}$IrO$_{3}$. Contrary to what has been believed so far, only for $x\leq0.25$ the system forms uniform solid solutions. For larger Li content, as evidenced by powder X-ray diffraction, scanning electron microscopy and density functional theory calculations, the system shows a miscibility gap and a phase separation into an ordered Na$_{3}$LiIr$_2$O$_{6}$ phase with alternating Na$_3$ and LiIr$_2$O$_6$ planes, and a Li-rich phase close to pure Li$_{2}$IrO$_{3}$. For $x\leq 0.25$ we observe (1) an increase of $c/a$ with Li doping up to $x=0.25$, despite the fact that $c/a$ in pure Li$_{2}$IrO$_{3}$ is smaller than in Na$_{2}$IrO$_{3}$, and (2) a gradual reduction of the antiferromagnetic ordering temperature $T_{N}$ and ordered moment. The previously proposed magnetic quantum phase transition at $x\approx 0.7$ may occur in a multiphase region and its nature needs to be re-evaluated.Comment: 8 pages, 9 figures including supplemental informatio

Average prime-pair counting formula

Taking $r>0$, let $\pi_{2r}(x)$ denote the number of prime pairs $(p, p+2r)$ with $p\le x$. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that $\pi_{2r}(x)\sim 2C_{2r} {\rm li}_2(x)$ with an explicit constant $C_{2r}>0$. There seems to be no good conjecture for the remainders $\omega_{2r}(x)=\pi_{2r}(x)- 2C_{2r} {\rm li}_2(x)$ that corresponds to Riemann's formula for $\pi(x)-{\rm li}(x)$. However, there is a heuristic approximate formula for averages of the remainders $\omega_{2r}(x)$ which is supported by numerical results.Comment: 26 pages, 6 figure

A new bound for the smallest xx with π(x)>li(x)\pi(x) > li(x)

We reduce the leading term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays and Hudson. Entering $2,000,000$ Riemann zeros, we prove that there exists $x$ in the interval $[exp(727.951858), exp(727.952178)]$ for which \pi(x)-\li(x) > 3.2 \times 10^{151}. There are at least $10^{154}$ successive integers $x$ in this interval for which \pi(x)>\li(x). This interval is strictly a sub-interval of the interval in Bays and Hudson, and is narrower by a factor of about 12.Comment: Final version, to be published in the International Journal of Number Theory [copyright World Scientific Publishing Company][www.worldscinet.com/ijnt

Efficient prime counting and the Chebyshev primes

The function \epsilon(x)=\mbox{li}(x)-\pi(x) is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions \epsilon_{\theta}(x)=\mbox{li}[\theta(x)]-\pi(x) and \epsilon_{\psi}(x)=\mbox{li}[\psi(x)]-\pi(x) are positive if and only if Riemann hypothesis (RH) holds (the first and the second Chebyshev function are $\theta(x)=\sum_{p \le x} \log p$ and $\psi(x)=\sum_{n=1}^x \Lambda(n)$, respectively, \mbox{li}(x) is the logarithmic integral, $\mu(n)$ and $\Lambda(n)$ are the M\"obius and the Von Mangoldt functions). Negative jumps in the above functions $\epsilon$, $\epsilon_{\theta}$ and $\epsilon_{\psi}$ may potentially occur only at $x+1 \in \mathcal{P}$ (the set of primes). One denotes j_p=\mbox{li}(p)-\mbox{li}(p-1) and one investigates the jumps $j_p$, $j_{\theta(p)}$ and $j_{\psi(p)}$. In particular, $j_p<1$, and $j_{\theta(p)}>1$ for $p<10^{11}$. Besides, $j_{\psi(p)}<1$ for any odd p \in \mathcal{\mbox{Ch}}, an infinite set of so-called {\it Chebyshev primes } with partial list $\{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, \ldots\}$. We establish a few properties of the set \mathcal{\mbox{Ch}}, give accurate approximations of the jump $j_{\psi(p)}$ and relate the derivation of \mbox{Ch} to the explicit Mangoldt formula for $\psi(x)$. In the context of RH, we introduce the so-called {\it Riemann primes} as champions of the function $\psi(p_n^l)-p_n^l$ (or of the function $\theta(p_n^l)-p_n^l$ ). Finally, we find a {\it good} prime counting function S_N(x)=\sum_{n=1}^N \frac{\mu(n)}{n}\mbox{li}[\psi(x)^{1/n}], that is found to be much better than the standard Riemann prime counting function.Comment: 15 pages section 2.2 added, new sequences added, Fig. 2 and 3 are ne

The Energetics of Li Off-Centering in K1−x_{1-x}Lix_xTaO3_3; First Principles Calculations

K$_{1-x}$Li$_{x}$TaO$_3$ (KLT) solid solutions exhibit a variety of interesting physical phenomena related to large displacements of Li-ions from ideal perovskite A-site positions. First-principles calculations for KLT supercells were used to investigate these phenomena. Lattice dynamics calculations for KLT exhibit a Li off-centering instability. The energetics of Li-displacements for isolated Li-ions and for Li-Li pairs up to 4th neighbors were calculated. Interactions between nearest neighbor Li-ions, in a Li-Li pair, strongly favor ferroelectric alignment along the pair axis. Such Li-Li pairs can be considered "seeds" for polar nanoclusters in KLT. Electrostriction, local oxygen relaxation, coupling to the KT soft-mode, and interactions with neighboring Li-ions all enhance the polarization from Li off-centering. Calculated hopping barriers for isolated Li-ions and for nearest neighbor Li-Li pairs are in good agreement with Arrhenius fits to experimental dielectric data.Comment: 14 pages including 10 figures. To Physical Review B. Replaced after corrections due to referees' remark