Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function

Let Δ(x) and E(t) denote respectively the remainder terms in the Dirichlet divisor problem and the mean square formula for the Riemann zeta-function on the critical line.This article is a survey of recent developments on the research of these famous error terms in number theory.These include upper bounds,Ω-results,sign changes,moments and distribution,etc.A few open problems are also discussed.     

Time correlation functions in classical dense fluids

The velocity autocorrelation functions and memory functions of dense classical fluids may be directly obtained from the static radial distribution function g(r) in an approximate way. Following the Mori projection operator formalism, the memory functions may be related to the fluctuating force correlation. At low densities, these functions may be evaluated by following the trajectories of particle pairs in the interatomic potential. At higher densities, the force correlation functions can be evaluated approximately from particle pair trajectories via the potential of the mean force. The contributions to the memory function come mainly from particle pairs with rather specific and rather short interatomic distances. At higher temperatures, this specific distance is even shorter, hence the memory function decays quickly with time. At lower temperatures, a negative region of the memory function may develop. On the other hand, there is relatively little density dependence of the normalized memory function. The results for argon fluids at various densities and temperatures agree satisfactorily with the molecular dynamics and the Enskog values. The decrease of the diffusion coefficient with density is partly due to the nature of g(r) which reflects the stronger clustering of atoms at higher densities.     

Electrochemically fabricated nanovolcano arrays for SERS applications

Nanovolcano arrays were fabricated by electrodepositing Ni/Cu alloys into a monolayer of self‐assembled nanospheres and then electrochemically etching the deposited alloy film. The fabricated nanovolcano arrays feature highly ordered hexagonally arranged concave nanobowls decorated with triangular nanopores at their interstices. After coated with Ag, the nanovolcano arrays serve as high‐performance substrates for surface enhancement Raman spectroscopy (SERS) measurements. The experimental study shows that the structural features of the nanovolcano arrays, including concave nanobowls, nanopores, and their long‐range orders, all contribute to the observed strong SERS enhancement in a synergetic manner, which is further confirmed by the simulation results obtained using the finite‐difference time‐domain method. Copyright © 2012 John Wiley & Sons, Ltd.     

Efficient approximately 2 microm Tm3+ -doped tellurite fiber laser

Laser emission in the range of 1.88-1.99 micrim from a Tm3+ -doped tellurite fiber is demonstrated when pumped with a diode-pumped Er3+/Yb3+-doped silica fiber laser operating at 1.57-1.61 microm. This pump source excites the Tm3+ ions directly into the F43 upper laser level and yields an output power of 280 mW with a slope efficiency of 76% in a 99%-12% laser cavity arrangement and a 32 cm long fiber. This result is very close to the Stokes efficiency limit of approximately 80%. This is, to the authors' knowledge, the first demonstration of high efficiency lasing in a tellurite fiber at wavelengths longer than 1.56 microm.     

Tm(3+)/Ho(3+) codoped tellurite fiber laser

Continuous-wave and Q-switched lasing from a Tm(3+)/Ho(3+) codoped tellurite fiber is reported. An Yb(3+)/Er(3+)-doped silica fiber laser operating at 1.6 microm was used as an in-band pump source, exciting the Tm(3+) ions into the (3)F(4) level. Energy is then nonradiatively transferred to the upper laser level, the (5)I(7) state of Ho(3+). The laser transition is from the (5)I(7) level to the (5)I(8) level, and the resulting emission is at 2.1 microm. For continuous wave operation, the slope efficiency was 62% and the threshold 0.1 W; the maximum output demonstrated was 0.16 W. Mechanical Q switching resulted in a pulse of 0.65 microJ energy and 160 ns duration at a repetition rate of 19.4 kHz.     

Record-high near-band-edge optical nonlinearities and two-level model correction of poled polymers by spectroscopic electromodulation and ellipsometry

The determination of nonlinearities near the band edge of organic and polymeric electro-optic(EO)materials is important from the viewpoint of molecular nonlinear optics(NLO)and photonic device applications.Based on transmission-mode Stark effect electromodulation(EM)spectroscopy,we study the electric-field-induced changes in optical absorption and refraction of newly developed EO polymers from the visible to near-infrared(NIR)wavelengths and report record-high near-band-edge complex EO effects from poled thin films.Values ofΔn andΔk up to 10^-3 and 10^-2 are found at an applied electric field of 2.0×10⁵-3.0×10⁵V/cm.The study of linear optical properties of poled films by spectroscopic ellipsometry shows large polinginduced birefringence and a nearly two-fold increase in the extinction coefficients at the extraordinary polarization.Through the Kramers-Kronig analysis,we obtained the real and imaginary second-order nonlinear coefficients up to~3,500 and~5,600 pm/V,respectively,which are believed to be the highest NLO coefficients of poled polymers through the resonance enhancement.Our approach goes beyond the previous works,applicable only to several discrete wavelengths,to a full-spectral analysis with independent verification of slab waveguide measurements.By considering both the electroabsorption and electrorefraction effects,our study overcomes the limitation of the classic qualitative two-level model and provides a quantitative understanding of near-resonance optical nonlinearities of organic EO materials.It can inspire the exploration of high-speed,absorptive,or phase-shifting light-modulators using EO polymers for on-chip applications.     

One step synthesis of Fe4.4Ni17.6Se16 coupled NiSe foam as self-supported,highly efficient and durable oxygen evolution electrode

The development of cost-effective, durable and high-efficient oxygen evolution reaction (OER) electrocatalysts is an extremely critical technology for the large-scale industrial water electrolysis. Here, a new strategy is proposed to significantly enhance the electrocatalytic activity by forming a hybrid electrode of NiSe and Fe_4.4Ni_17.6Se₁₆ through direct selenization of porous iron-nickel (FeNi) alloy foam via thermal selenization process. The obtained self-supported Fe_4.4Ni_17.6Se₁₆/NiSe hybrid (FNS/NiSe) foam displays outstanding durability and remarkable catalytic activity in 1.0 M KOH with low overpotentials of 242 and 282 mV to achieve the current densities of 100 and 500 mA cm^?2, respectively. To the best of our knowledge, it exceeds most of the reported OER electrocatalysts in alkaline electrolytes.     

Orbital-Overlap control in the solid-state reactivity of beta-azido-propiophenones: selective formation of cis-azo-dimers

Solid-state photolysis of 1a,b yields selectively cis-3a,b. X-ray analysis of 1a,b reveals the molecules adopt an extended structure and as such the crystal packing arrangement consists of planar, pi-stacked molecules. The shortest intermolecular distance between adjacent N-atoms is approximately 3.76 A and would lead to formation of trans-3a,b, whereas cis-3a,b is formed by dimerization between N-atoms that are approximately 3.9 A apart. We propose that the molecular orbital alignment of the adjacent nitrenes controls the solid-state reactivity.     

Stereoselective isomerisation of N-allyl aziridines into geometrically stable Z enamines by using rhodium hydride catalysis

In the presence of rhodium(I) hydride catalysts, tertiary N-allylamines are known to isomerise into E enamines. In contrast, we have recently found that N-allylaziridines isomerise to form Z enamines. On the basis of literature data, the most likely mechanism of isomerisation would involve a rhodium hydride addition/beta-hydride elimination sequence. We show that the observed selectivity cannot be adequately explained by this pathway and is more consistent with initial CH-activation followed by rearrangement to form a five-membered cyclometallated rhodium intermediate. This intermediate subsequently undergoes reductive elimination to form a C--H bond. The resulting geometrically stable Z enamines are useful building blocks for stereoselective synthesis.