Dense clusters of primes in subsets

We prove a generalization of the author's work to show that any subset of the primes which is `well-distributed' in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the parameters. As applications, we show there are infinitely many intervals of length $(\log{x})^{\epsilon}$ containing $\gg_\epsilon \log\log{x}$ primes, and show lower bounds of the correct order of magnitude for the number of strings of $m$ congruent primes with $p_{n+m}-p_n\le \epsilon\log{x}$.Comment: 35 pages; clarified some statement

Primes represented by incomplete norm forms

Let $K=\mathbb{Q}(\omega)$ with $\omega$ the root of a degree $n$ monic irreducible polynomial $f\in\mathbb{Z}[X]$. We show the degree $n$ polynomial $N(\sum_{i=1}^{n-k}x_i\omega^{i-1})$ in $n-k$ variables formed by setting the final $k$ coefficients to 0 takes the expected asymptotic number of prime values if $n\ge 4k$. In the special case $K=\mathbb{Q}(\sqrt[n]{\theta})$, we show $N(\sum_{i=1}^{n-k}x_i\sqrt[n]{\theta^{i-1}})$ takes infinitely many prime values provided $n\ge 22k/7$. Our proof relies on using suitable `Type I' and `Type II' estimates in Harman's sieve, which are established in a similar overall manner to the previous work of Friedlander and Iwaniec on prime values of $X^2+Y^4$ and of Heath-Brown on $X^3+2Y^3$. Our proof ultimately relies on employing explicit elementary estimates from the geometry of numbers and algebraic geometry to control the number of highly skewed lattices appearing in our final estimates.Comment: 103 pages; v2 is significant rewrite of v1, main results unchange

Sieve weights and their smoothings

We obtain asymptotic formulas for the $2k$th moments of partially smoothed divisor sums of the M\"obius function. When $2k$ is small compared with $A$, the level of smoothing, then the main contribution to the moments come from integers with only large prime factors, as one would hope for in sieve weights. However if $2k$ is any larger, compared with $A$, then the main contribution to the moments come from integers with quite a few prime factors, which is not the intention when designing sieve weights. The threshold for "small" occurs when $A=\frac 1{2k} \binom{2k}{k}-1$. One can ask analogous questions for polynomials over finite fields and for permutations, and in these cases the moments behave rather differently, with even less cancellation in the divisor sums. We give, we hope, a plausible explanation for this phenomenon, by studying the analogous sums for Dirichlet characters, and obtaining each type of behaviour depending on whether or not the character is "exceptional".Comment: Final version, 85 pages, to appear in Ann. Sci. \'Ec. Norm. Sup\'er.. Added abstract in French and made several minor changes compared to the previous versio

The Need for an Independent Federal Judiciary: Prison Reform Litigation Spurs Structural Change – The California Penal Crisis

This Article examines Madrid V. Gomez, 889 F.Supp. 1146, the last case before the passage of the PLRA in which a federal court broadly intervened in a state prison system through structural reform litigation. Part II outlines the historical and jurisprudential foundations that legitimate federal judicial intervention in state prisons. Part III examines the California prison system through the lens of the Madrid litigation and the ongoing social and political problems caused by the prison crisis. Part IV concludes that judicial intervention remains the only viable tool to remedy constitutional deficiencies in state prisons when majoritarian political processes fail to produce serious reform. Therefore, it is essential that the power and independence of the federal judiciary be preserved to ensure the rights of the politically powerless and mitigate Constitutional harms

Long gaps in sieved sets

For each prime $p$, let $I_p \subset \mathbb{Z}/p\mathbb{Z}$ denote a collection of residue classes modulo $p$ such that the cardinalities $|I_p|$ are bounded and about $1$ on average. We show that for sufficiently large $x$, the sifted set $\{ n \in \mathbb{Z}: n \pmod{p} \not \in I_p \hbox{ for all }p \leq x\}$ contains gaps of size at least $x (\log x)^{\delta}$ where $\delta>0$ depends only on the density of primes for which $I_p\ne \emptyset$. This improves on the ``trivial'' bound of $\gg x$. As a consequence, for any non-constant polynomial $f:\mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient, the set $\{ n \leq X: f(n) \hbox{ composite}\}$ contains an interval of consecutive integers of length $\ge (\log X) (\log\log X)^{\delta}$ for sufficiently large $X$, where $\delta>0$ depends only on the degree of $f$.Comment: Major revision. We replaced the PNT-type assumption with (a) a Mertens estimate; (b) that the density $\rho$ of nonempty $I_p$ exists. Our main theorem now gives an exponent which is a function of $\rho$, and is completely explicit. In particular, the exponent $e^{-1-4/\rho}$ is admissible. Various notational simplifications. Many remarks added to help the reade

On limit points of the sequence of normalized prime gaps

Let $p_n$ denote the $n$th smallest prime number, and let $\boldsymbol{L}$ denote the set of limit points of the sequence $\{(p_{n+1} - p_n)/\log p_n\}_{n = 1}^{\infty}$ of normalized differences between consecutive primes. We show that for $k = 9$ and for any sequence of $k$ nonnegative real numbers $\beta_1 \le \beta_2 \le ... \le \beta_k$, at least one of the numbers $\beta_j - \beta_i$ ($1 \le i < j \le k$) belongs to $\boldsymbol{L}$. It follows at least $12.5$ of all nonnegative real numbers belong to $\boldsymbol{L}$.Comment: Revised and improve

Preliminary evidence for the treatment of performance blocks in sport: The efficacy of EMDR With Graded Exposure

Sport psychologists are increasingly confronted with performance problems in sport where athletes suddenly lose the ability to execute automatic movements (Rotheram, Maynard, Thomas, Bawden, & Francis, 2012). Described as performance blocks (Bennett, Hays, Lindsay, Olusoga, & Maynard, 2015), these problems manifest as locked, stuck, and frozen movements and are underpinned by an aggressive anxiety component. This research used both qualitative and quantitative methods in a single case study design to investigate the effectiveness of eye movement desensitization and reprocessing (EMDR) therapy with graded exposure as a treatment method. The participant was a 58-year-old professional male golfer who had been suffering a performance block for 11 years. Specifically, the participant was experiencing involuntary spasms, shaking, muscle tension, and jerking in the lower left forearm while executing a putting stroke. Physical symptoms were coupled with extreme anxiety, panic, and frustration. The study tested the hypothesis that reprocessing related significant life events and attending to dysfunctional emotional symptoms would eliminate the performance block and related symptoms and that the individual would regain his ability to execute the affected skill. Pre-, mid-, and postintervention performance success, using the Impact of Event scale, subjective units of distress (SUD; Wolpe, 1973), and kinematic testing revealed improvements in all associated symptoms in training and competition. These findings suggest that previous life experiences might be associated with the onset of performance blocks and that EMDR with graded exposure might offer an effective treatment method