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

A study of the kinetics of the oxidative coupling of methane over a Li/Sn/MgO catalyst

The rate of reaction of methane with oxygen in the presence of a Li/Sn/MgO catalyst has been studied as a function of the partial pressures of CH4, O2 and CO2 using a well-mixed reaction system which is practically gradientless with respect to gas-phase concentrations. It is concluded that the rate-determining step involves reaction of a molecule of CH4 adsorbed on the catalyst surface with an adsorbed di-atomic oxygen species. The kinetics are consistent with a Langmuir-Hinshelwood type mechanism involving competitive adsorption of CH4, O2 and CO2 on a single site. A comparison is made with previously published results for the Li/MgO material

On the lithium content of the globular cluster M92

I use literature data and a new temperature calibration to determine the Li abundances in the globular cluster M 92. Based on the same data, Boesgaard et al. have claimed that there is a dispersion in Li abundances in excess of observational errors. This result has been brought as evidence for Li depletion in metal-poor dwarfs. In the present note I argue that there is no strong evidence for intrinsic dispersion in Li abundances, although a dispersion as large as 0.18 dex is possible. The mean Li abundance, A(Li)=2.36, is in good agreement with recent results for field stars and TO stars in the metal-poor globular cluster NGC 6397. The simplest interpretation is that this constant value represents the primordial Li abundance.Comment: A&A accepte

First-principles prediction of redox potentials in transition-metal compounds with LDA+U

First-principles calculations within the Local Density Approximation (LDA) or Generalized Gradient Approximation (GGA), though very successful, are known to underestimate redox potentials, such as those at which lithium intercalates in transition metal compounds. We argue that this inaccuracy is related to the lack of cancellation of electron self-interaction errors in LDA/GGA and can be improved by using the DFT+$U$ method with a self-consistent evaluation of the $U$ parameter. We show that, using this approach, the experimental lithium intercalation voltages of a number of transition metal compounds, including the olivine Li$_{x}$MPO$_{4}$ (M=Mn, Fe Co, Ni), layered Li$_{x}$MO$_{2}$ ($x=$Co, Ni) and spinel-like Li$_{x}$M$_{2}$O$_{4}$ (M=Mn, Co), can be reproduced accurately.Comment: 19 pages, 6 figures, Phys. Rev. B 70, 235121 (2004

Lithium distribution across the membrane of motoneurons in the isolated frog spinal cord

Lithium sensitive microelectrodes were used to investigate the transmembrane distribution of lithium ions (Li+) in motoneurons of the isolated frog spinal cord. After addition of 5 mmol·l–1 LiCl to the bathing solution the extracellular diffusion of Li+ was measured. At a depth of 500 m, about 60 min elapsed before the extracellular Li+ concentration approached that of the bathing solution. Intracellular measurements revealed that Li+ started to enter the cells soon after reaching the motoneuron pool and after up to 120 min superfusion, an intra — to extracellular concentration ratio of about 0.7 was obtained. The resting membrane potential and height of antidromically evoked action potentials were not altered by 5 mmol·l–1 Li+

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

Cerium: the lithium substitute in post-AGB stars

In this letter we present an alternative identification for the line detected in the spectra of s-process enriched low-mass post-AGB stars around 6708A and which was interpreted in the literature as due to Li. Newly released line lists of lanthanide species reveal, however, the likely identification of the line to be due to a CeII transition. We argue that this identification is consistent with the Ce abundance of all the objects discussed in the literature and conclude that in none of the low-mass s-process enriched post-AGB stars there is indication for Li-production.Comment: 5 pages, 2 figures, accepted for publication as A&A Lette

Constraints to the Masses of Brown Dwarf Candidates from the Lithium Test

We present intermediate dispersion (0.7-2.2 \AA ~pix$^{-1}$) optical spectroscopic observations aimed at applying the ``Lithium Test'' to a sample of ten brown dwarf candidates located in the general field, two in young open clusters, and two in close binaries. We find evidence for strong Li depletion in all of them, and thus infer lower mass limits of 0.065~M$_\odot$, depending only slightly ($\pm$0.005~M$_\odot$) on the interior models. None of the field brown dwarf candidates in our sample appears to be a very young (age $<$~10$^8$~yr) substellar object. For one of the faintest proper motion Pleiades members known (V=20.7) the Li test implies a mass greater than $\sim$0.08~M$_\odot$, and therefore it is not a brown dwarf. From our spectra we estimate spectral types for some objects and present measurements of Halpha emission strengths and radial velocities. Finally, we compare the positions in the H-R diagram of our sample of brown dwarf candidates with the theoretical region where Li is expected to be preserved (Substellar Lithium Region). We find that certain combinations of temperature calibrations and evolutionary tracks are consistent with the constraints imposed by the observed Li depletion in brown dwarf candidates, while others are not.Comment: 20 pp.; 4 figs, available under request; plain LaTeX, ApJ in press, OACatania-94-00