Let $mathcal{R}$ be a finite set of integers satisfying appropriate local conditions. We show the existence of long clusters of primes $p$ in bounded length intervals with $p-b$ squarefree for all $b in mathcal{R}$. Moreover, we can enforce that the
primes $p$ in our cluster satisfy any one of the following conditions: (1) $p$ lies in a short interval $[N, N+N^{frac{7}{12}+epsilon}]$, (2) $p$ belongs to a given inhomogeneous Beatty sequence, (3) with $c in (frac{8}{9},1)$ fixed, $p^c$ lies in a prescribed interval mod $1$ of length $p^{-1+c+epsilon}$.
By combining a sieve method of Harman with the work of Maynard and Tao we show that $$liminf_{nrightarrow infty}(p_{n+m}-p_n)ll exp(3.815m).$$
Vibrating nano- and micromechanical resonators have been the subject of research aiming at ultrasensitive mass sensors for mass spectrometry, chemical analysis and biomedical diagnosis. Unfortunately, their merits diminish dramatically in liquids due
to dissipative mechanisms like viscosity and acoustic losses. A push towards faster and lighter miniaturized nanodevices would enable improved performances, provided dissipation was controlled and novel techniques were available to efficiently drive and read-out their minute displacement. Here we report on a nano-optomechanical approach to this problem using miniature semiconductor disks. These devices combine mechanical motion at high frequency above the GHz, ultra-low mass of a few picograms, and moderate dissipation in liquids. We show that high-sensitivity optical measurements allow to direct resolve their thermally driven Brownian vibrations, even in the most dissipative liquids. Thanks to this novel technique, we experimentally, numerically and analytically investigate the interaction of these resonators with arbitrary liquids. Nano-optomechanical disks emerge as probes of rheological information of unprecedented sensitivity and speed, opening applications in sensing and fundamental science.
Let $t in mathbb{N}$, $eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q leq x^{5/12-eta}$, $q$ not a multiple of the conductor of the exceptional character $chi^*$ (if it exists). Suppose further that,
[ max {p : p | q } < exp (frac{log x}{C log log x}) ; ; {and} ; ; prod_{p | q} p < x^{delta}, ] where $C$ and $delta$ are suitable positive constants depending on $t$ and $eta$. Let $a in mathbb{Z}$, $(a,q)=1$ and [ mathcal{A} = {n in (x/2, x]: n equiv a pmod{q} } . ] We prove that there are primes $p_1 < p_2 < ... < p_t$ in $mathcal{A}$ with [ p_t - p_1 ll qt exp (frac{40 t}{9-20 theta}) . ] Here $theta = (log q) / log x$.
We report on miniature GaAs disk optomechanical resonators vibrating in air in the radiofrequency range. The flexural modes of the disks are studied by scanning electron microscopy and optical interferometry, and correctly modeled with the elasticity
theory for annular plates. The mechanical damping is systematically measured, and confronted with original analytical models for air damping. Formulas are derived that correctly reproduce both the mechanical modes and the damping behavior, and can serve as design tools for optomechanical applications in fluidic environment.
