Turing computability, probability, and prime numbers

We present an original theoretical approach to prove that $\pi (n)-Li(n)=o(M(n)\sqrt{Li(n)})$ almost certainly stands, where $\pi (n)$ is the number of primes not greater than $n$, $Li(n)$ is a logarithmic integral function, and $M(n)$ is an arbitrary function such that $M(n)\rightarrow\infty$.Comment: Revision of the contents over the whole range of the pape

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

Quasiparticle electronic band structure of the alkali metal chalcogenides

The electronic energy band spectra of the alkali metal chalcogenides M$_2$A (M: Li, Na, K, Rb; A: O, S, Se, Te) have been evaluated within the projector augmented waves (PAW) approach by means of the ABINIT code. The Kohn-Sham single-particle states have been found in the GGA (the generalized gradient approximation) framework. Further, on the basis of these results the quasiparticle energies of electrons as well as the dielectric constants were obtained in the GW approximation. The calculations based on the Green's function have been originally done for all the considered M$_2$A crystals, except Li$_2$O.Comment: 8 pages, 8 figure

Lithium in the Intermediate-Age Open Cluster, NGC 3680

High-dispersion spectra centered on the Li 6708 A line have been obtained for 70 potential members of the open cluster NGC 3680, with an emphasis on stars in the turnoff region. A measurable Li abundance has been derived for 53 stars, 39 of which have radial velocities and proper motions consistent with cluster membership. After being transferred to common temperature and abundance scales, previous Li estimates have been combined to generate a sample of 49 members, 40 of which bracket the cluster Li-dip. Spectroscopic elemental analysis of 8 giants and 5 turnoff stars produces [Fe/H] = -0.17 +/- 0.07 (sd) and -0.07 +/- 0.02 (sd), respectively. We also report measurements of Ca, Si and Ni which are consistent with scaled-solar ratios within the errors. Adopting [Fe/H] = -0.08 (Sect. 3.6), Y^2 isochrone comparisons lead to an age of 1.75 +/- 0.10 Gyr and an apparent modulus of (m-M) = 10.30 +/- 0.15 for the cluster, placing the center of the Li-dip at 1.35 +/- 0.03 solar masses. Among the giants, 5 of 9 cluster members are now known to have measurable Li with A(Li) near 1.0. A combined sample of dwarfs in the Hyades and Praesepe is used to delineate the Li-dip profile at 0.7 Gyr and [Fe/H] = +0.15, establishing its center at 1.42 +/- 0.02 solar masses and noting the possible existence of secondary dip on its red boundary. When evolved to the typical age of the clusters NGC 752, IC 4651 and NGC 3680, the Hyades/Praesepe Li-dip profile reproduces the observed morphology of the combined Li-dip within the CMD's of the intermediate-age clusters while implying a metallicity dependence for the central mass of the Li-dip given by Mass = (1.38 +/-0.04) + (0.4 +/- 0.2)[Fe/H]. The implications of the similarity of the Li-dichotomy among giants in NGC 752 and IC 4651 and the disagreement with the pattern among NGC 3680 giants are discussed.Comment: Latex ms. is 56 pages, including 10 figures and 4 tables. Accepted for the Astronomical Journa

The Median Largest Prime Factor

Let $M(x)$ denote the median largest prime factor of the integers in the interval $[1,x]$. We prove that $$M(x)=x^{\frac{1}{\sqrt{e}}\exp(-\text{li}_{f}(x)/x)}+O_{\epsilon}(x^{\frac{1}{\sqrt{e}}}e^{-c(\log x)^{3/5-\epsilon}})$$ where $\text{li}_{f}(x)=\int_{2}^{x}\frac{\{x/t\}}{\log t}dt$. From this, we obtain the asymptotic $$M(x)=e^{\frac{\gamma-1}{\sqrt{e}}}x^{\frac{1}{\sqrt{e}}}(1+O(\frac{1}{\log x})),$$ where $\gamma$ is the Euler Mascheroni constant. This answers a question posed by Martin, and improves a result of Selfridge and Wunderlich.Comment: 7 page

Nonflammable Lithium Metal Full Cells with Ultra-high Energy Density Based on Coordinated Carbonate Electrolytes

Coupling thin Li metal anodes with high-capacity/high-voltage cathodes such as LiNi0.8Co0.1Mn0.1O2 (NCM811) is a promising way to increase lithium battery energy density. Yet, the realization of high-performance full cells remains a formidable challenge. Here, we demonstrate a new class of highly coordinated, nonflammable carbonate electrolytes based on lithium bis(fluorosulfonyl)imide (UFSI) in propylene carbonate/fluoroethylene carbonate mixtures. Utilizing an optimal salt concentr ation (4 M LiFSI) of the electrolyte results in a unique coordination structure of Li+-FSI-solvent cluster, which is critical for enabling the formation of stable interfaces on both the thin Li metal anode and high-voltage NCM811 cathode. Under highly demanding cell configuration and operating conditions (Li metal anode = 35 mu m, areal capacity/charge voltage of NCM811 cathode = 4.8 mAh cm(-2)/4 .6 V, and anode excess capacity [relative to the cathode] = 0.83), the Li metal-based full cell provides exceptional electrochemical performance (energy densities = 679 Wh kg(cell)(-1)/1,024 Wh L-cell(-1)) coupled with nonflammability

Magnetic and Electronic Properties of Lix_xCoO2_2 Single Crystals

Measurements of electrical resistivity ($\rho$), DC magnetization ($M$) and specific heat ($C$) have been performed on layered oxide Li$_x$CoO$_2$ (0.25$\leq$$x$$\leq$0.99) using single crystal specimens. The $\rho$ versus temperature ($T$) curve for $x$=0.90 and 0.99 is found to be insulating but a metallic behavior is observed for 0.25$\leq$$x$$\leq$0.71. At $T_{\rm S}$$\sim$155 K, a sharp anomaly is observed in the $\rho$$-$$T$, $M$$-$$T$ and $C$$/$$T$$-$$T$ curves for $x$=0.66 with thermal hysteresis, indicating the first-order charactor of the transition. The transition at $T_{\rm S}$$\sim$155 K is observed for the wide range of $x$=0.46$-$0.71. It is found that the $M$$-$$T$ curve measured after rapid cool becomes different from that after slow cool below $T_{\rm F}$, which is $\sim$130 K for $x$=0.46$-$0.71. $T_{\rm F}$ is found to agree with the temperature at which the motional narrowing in the $^7$Li NMR line width is observed, indicating that the Li ions stop diffusing and order at the regular site below $T_{\rm F}$. The ordering of Li ions below $T_{\rm F}$$\sim$130 K is likely to be triggered and stabilized by the charge ordering in CoO$_2$ layers below $T_{\rm S}$.Comment: 8 pages, 7 figure