Prime numbers in logarithmic intervals

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and h=\odi{X}. Then we will apply this to prove that for every $\lambda>1/2$ there exists a positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda\log X]$ contains at least a prime number. As a consequence we improve Cheer and Goldston's result on the size of real numbers $\lambda>1$ with the property that there is a positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda\log X]$ contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda\log X]$ contains at least a prime number. The last application of these techniques are two theorems (the first one unconditional and the second one in which we assume the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture) on the positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda\log X]$ contains no primes.Comment: 17 page

Città della Conciliazione, Grugliasco, Torino, Italy, 2005-2009

Pubblicazione del progetto Città Universitaria della Conciliazione a Grugliasco (TO) nel numero della rivista World Architecture dedicato ai progetti della Città di Torino

Città Universitaria della Conciliazione a Grugliasco (TO)

Pubblicazione e testo critico del progetto della Città Universitaria della Conciliazione contenuto nell'articolo di Davide Tommaso Ferrando," A welcoming space. Sustainability as conciliation between work and family life" nel numero della rivista World Architecture dedicato ai progetti della Città di Torino

The exceptional set for the number of primes in short intervals

We give upper bounds for the number of x up to X such that the interval (x, x+h) does not contain the expected quantity of primes. Here h is small with respect to x

Città Universitaria della Conciliazione a Grugliasco

Pubblicazione e commento critico del progetto della Città Universitaria della Conciliazione contenuto nell'editoriale di Zhang Li (vice-direttore) "Soft sustainability: the Torino approach" del numero della rivista World Architecture dedicato alla Città di Torino

On the thermodynamic path enabling a room-temperature, laser-assisted graphite to nanodiamond transformation

Nanodiamonds are the subject of active research for their potential applications in nano-magnetometry, quantum optics, bioimaging and water cleaning processes. Here, we present a novel thermodynamic model that describes a graphite-liquid-diamond route for the synthesis of nanodiamonds. Its robustness is proved via the production of nanodiamonds powders at room-temperature and standard atmospheric pressure by pulsed laser ablation of pyrolytic graphite in water. The aqueous environment provides a confinement mechanism that promotes diamond nucleation and growth, and a biologically compatible medium for suspension of nanodiamonds. Moreover, we introduce a facile physico-chemical method that does not require harsh chemical or temperature conditions to remove the graphitic byproducts of the laser ablation process. A full characterization of the nanodiamonds by electron and Raman spectroscopies is reported. Our model is also corroborated by comparison with experimental data from the literature

On the asymptotic formula for Goldbach numbers in short intervals

Let $R(k)=\sum\limits_{l+m=k}\Lambda(l)\Lambda(m)$, \Sing(k) = 2 \prod\limits_{p>2}\left(1-\frac{1}{(p-1)^2}\right) \prod\limits_{\substack{ p\mid k\\ p>2 }} \left(\frac{p-1}{p-2}\right) if $k$ is even and \Sing(k) =0 if $k$ is odd. It is known that R(k) \sim k\Sing(k) as $N\to \infty$ for almost all $k\in [N,2N]$ and that \sum_{k\in [n,n+H)}R(k) \sim \sum_{k\in [n,n+H)} k\Sing(k) \quad\hbox{for} \quad n\to \infty \eqno{(1)} uniformly for $H\geq n^{1/6+\epsilon}$. Here we prove, assuming $N^\epsilon\leq H\leq N^{1/6+\epsilon}$ and $N\to\infty$, that (1) holds for almost all $n\in [N,2N]$