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 Synthesis and Characterization of New, Robust Titanium (IV) Scorpionate Complexes

Titanium complexes possessing sterically encumbered ligands have allowed for the preparation of reactive moieties (imido, alkylidene and alkylidyne species) relevant to reactions such as olefin polymerization and alkyne hydroamination. For this reason, we have targeted robust scorpionate ancillary ligands to support reactive titanium centers. Thus, a series of titanium complexes were synthesized using an achiral oxazoline-based scorpionate ligand, tris(4,4-dimethyl-2-oxazolinyl)phenyl borate [To^M^]^-^ as well as the related chiral ligand, tris(4-isopropyl-2-oxazolinyl)phenyl borate [To^P^]^-^. The complex [Ti(&#x3ba;^3^- To^M^)Cl~3~] was prepared in moderate yield (43%) by the rapid (&#x3c;1 min at room temperature) reaction of Li[To^M^] and TiCl~4~ in methylene chloride; this new compound was characterized by ^1^H NMR spectroscopy as the expected C~3v~-symmetric species. One route to Ti (IV) alkyls involves salt metathesis; accordingly, syntheses of [To^M^]Ti alkyl complexes by interaction of [Ti(&#x3ba;^3^-To^M^)Cl~3~] and one or three equivalents of alkylating agents, such as benzyl potassium (KCH~2~C~6~H~5~), trimethylsilylmethyl&#xd;&#xa;lithium (LiCH~2~Si(CH~3~) ~3~), or neopentyl lithium (LiCH~2~C(CH~3~)~3~) are currently under investigation. The complexes [Ti(=NBut) (&#x3ba;~3~-To^M^)(Cl)(Bu^t^py)] (Bu^t^py=4 tert-butylpyridine) and [Ti(=NBu^t^) (&#x3ba;~3~-To^P^)(Cl)(Bu^t^py)] were synthesized by reaction of the known Ti imido [Ti(=NBu^t^)(Cl)~2~(Bu^t^py)~2~] with Li[To^M^] or Li[To^P^], respectively, by stirring overnight in methylene chloride at ambient temperature. The complexes were identified using ^1^H NMR spectroscopy, ^1^H-^13^C HMQC and ^1^H-^15^N HMBC correlation experiments

Ferrocenyl hydroxymethylphosphines (η⁵-C₅H₅)Fe[η⁵⁻C₅H₄P(CH₂OH)₂] and 1,1′-[Fe{η⁵-C₅H₄P(CH₂OH)₂}₂] and their chalcogenide derivatives

The ferrocenyl hydroxymethylphosphines FcP(CH₂OH)₂ [Fc=(η⁵-C₅H₅)Fe(η⁵-C₅H₄)] and 1,1′-Fc′[P(CH₂OH)₂]₂ [Fc′=Fe(η⁵⁻C₅H₄)₂] were prepared by reactions of the corresponding primary phosphines FcPH₂ and 1,1′-Fc′(PH₂)₂ with excess aqueous formaldehyde. The crystal structure of FcP(CH₂OH)₂ was determined and compared with the known ferrocenyl hydroxymethylphosphine FcCH₂P(CH₂OH)₂. The chalcogenide derivatives FcP(E)(CH₂OH)₂ and 1,1′-Fc′[P(E)(CH₂OH)₂]₂ (E=O, S, Se) were prepared and fully characterised. Crystal structure determinations on FcP(O)(CH₂OH)₂ and FcP(S)(CH₂OH)₂ were performed, and the hydrogen-bonding patterns are compared with related compounds. The sulfide shows no hydrogen-bonding involving the phosphine sulfide group, in contrast to other reported ferrocenyl hydroxymethylphosphine sulfides. The platinum complex cis-[PtCl₂{FcP(CH₂OH)₂}₂] was prepared by reaction of 2 mol equivalents of FcP(CH₂OH)₂ with [PtCl₂(1,5-cyclo-octadiene)], and was characterised by 31P-NMR spectroscopy and negative ion electrospray mass spectrometry, which gave a strong [M+Cl]⁻ ion

Feshbach resonances in ultracold ^{6,7}Li + ^{23}Na atomic mixtures

We report a theoretical study of Feshbach resonances in $^{6}$Li + $^{23}$Na and $^{7}$Li + $^{23}$Na mixtures at ultracold temperatures using new accurate interaction potentials in a full quantum coupled-channel calculation. Feshbach resonances for $l=0$ in the initial collisional open channel $^6$Li$(f=1/2, m_f=1/2) + ^{23}$Na$(f=1, m_f=1)$ are found to agree with previous measurements, leading to precise values of the singlet and triplet scattering lengths for the $^{6,7}$Li$+^{23}$Na pairs. We also predict additional Feshbach resonances within experimentally attainable magnetic fields for other collision channels.Comment: 4 pages, 3 figure

Feshbach resonances in mixtures of ultracold 6^6Li and 87^{87}Rb gases

We report on the observation of two Feshbach resonances in collisions between ultracold $^6$Li and $^{87}$Rb atoms in their respective hyperfine ground states $|F,m_F>=|1/2,1/2>$ and $|1,1>$. The resonances show up as trap losses for the $^6$Li cloud induced by inelastic Li-Rb-Rb three-body collisions. The magnetic field values where they occur represent important benchmarks for an accurate determination of the interspecies interaction potentials. A broad Feshbach resonance located at 1066.92 G opens interesting prospects for the creation of ultracold heteronuclear molecules. We furthermore observe a strong enhancement of the narrow p-wave Feshbach resonance in collisions of $^6$Li atoms at 158.55 G in the presence of a dense $^{87}$Rb cloud. The effect of the $^{87}$Rb cloud is to introduce Li-Li-Rb three-body collisions occurring at a higher rate than Li-Li-Li collisions.Comment: 4 pages, 3 figure

Analytical Approximations for the Collapse of an Empty Spherical Bubble

The Rayleigh equation 3/2 R'+RR"+p/rho=0 with initial conditions R(0)=Rmax, R'(0)=0 models the collapse of an empty spherical bubble of radius R(T) in an ideal, infinite liquid with far-field pressure p and density rho. The solution for r=R/Rmax as a function of time t=T/Tcollapse, where R(Tcollapse)=0, is independent of Rmax, p, and rho. While no closed-form expression for r(t) is known we find that s(t)=(1-t^2)^(2/5) approximates r(t) with an error below 1%. A systematic development in orders of t^2 further yields the 0.001%-approximation r*(t)=s(t)[1-a Li(2.21,t^2)], where a=-0.01832099 is a constant and Li is the polylogarithm. The usefulness of these approximations is demonstrated by comparison to high-precision cavitation data obtained in microgravity.Comment: 5 pages, 2 figure

An Extremely Lithium-Rich Bright Red Giant in the Globular Cluster M3

We have serendipitously discovered an extremely lithium-rich star on the red giant branch of the globular cluster M3 (NGC 5272). An echelle spectrum obtained with the Keck I HIRES reveals a Li I 6707 Angstrom resonance doublet of 520 milli-Angstrom equivalent width, and our analysis places the star among the most Li-rich giants known: log[epsilon(Li)] ~= +3.0. We determine the elemental abundances of this star, IV-101, and three other cluster members of similar luminosity and color, and conclude that IV-101 has abundance ratios typical of giants in M3 and M13 that have undergone significant mixing. We discuss mechanisms by which a low-mass star may be so enriched in Li, focusing on the mixing of material processed by the hydrogen-burning shell just below the convective envelope. While such enrichment could conceivably only happen rarely, it may in fact regularly occur during giant-branch evolution but be rarely detected because of rapid subsequent Li depletion.Comment: 7-page LaTeX file, including 2 encapsulated ps figures + 1 table; accepted for publication in the Astrophysical Journal Letter