On rr-gaps between zeros of the Riemann zeta-function

Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) > r + \Theta\sqrt{r}$ and $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) < r - \vartheta\sqrt{r}$ for explicit absolute positive constants $\Theta$ and $\vartheta$, where $\gamma$ denotes an ordinate of a zero of the Riemann zeta-function on the critical line. Selberg published announcements of this result several times but did not include a proof. We also suggest a general framework which might lead to stronger statements concerning the vertical distribution of nontrivial zeros of the Riemann zeta-function.Comment: to appear in the Bulletin of the London Mathematical Societ

Consecutive primes in tuples

In a recent advance towards the Prime $k$-tuple Conjecture, Maynard and Tao have shown that if $k$ is sufficiently large in terms of $m$, then for an admissible $k$-tuple $\mathcal{H}(x) = \{gx + h_j\}_{j=1}^k$ of linear forms in $\mathbb{Z}[x]$, the set $\mathcal{H}(n) = \{gn + h_j\}_{j=1}^k$ contains at least $m$ primes for infinitely many $n \in \mathbb{N}$. In this note, we deduce that $\mathcal{H}(n) = \{gn + h_j\}_{j=1}^k$ contains at least $m$ consecutive primes for infinitely many $n \in \mathbb{N}$. We answer an old question of Erd\H os and Tur\'an by producing strings of $m + 1$ consecutive primes whose successive gaps $\delta_1,\ldots,\delta_m$ form an increasing (resp. decreasing) sequence. We also show that such strings exist with $\delta_{j-1} \mid \delta_j$ for $2 \le j \le m$. For any coprime integers $a$ and $D$ we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class $a \bmod D$.Comment: Revised versio

An effective Chebotarev density theorem for families of number fields, with an application to ℓ\ell-torsion in class groups

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree.Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.0200

Extremal primes for elliptic curves without complex multiplication

Fix an elliptic curve E over Q. An extremal prime for E is a prime p of good reduction such that the number of rational points on E modulo p is maximal or minimal in relation to the Hasse bound. Assuming that all the symmetric power L-functions associated to E are automorphic and satisfy the Generalized Riemann Hypothesis, we give the first non-trivial upper bounds for the number of such primes when E is a curve without complex multiplication. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in the work of Rouse and Thorner (arXiv:1305.5283) and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure

Benford Behavior of Zeckendorf Decompositions

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density of the elements in $S$ with leading digit $d$ is $\log_{10}{(1+\frac{1}{d})}$; in other words, smaller leading digits are more likely to occur. We prove that, as $n\to\infty$, for a randomly selected integer $m$ in $[0, F_{n+1})$ the distribution of the leading digits of the Fibonacci summands in its Zeckendorf decomposition converge to Benford's law almost surely. Our results hold more generally, and instead of looking at the distribution of leading digits one obtains similar theorems concerning how often values in sets with density are attained.Comment: Version 1.0, 12 pages, 1 figur

Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals

Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers ${F_n}$, with initial terms $F_1 = 1, F_2 = 2$. We consider the distribution of the number of summands involved in such decompositions. Previous work proved that as $n \to \infty$ the distribution of the number of summands in the Zeckendorf decompositions of $m \in [F_n, F_{n+1})$, appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in $[F_n, F_{n+1})$ share the same potential summands. We generalize these results to subintervals of $[F_n, F_{n+1})$ as $n \to \infty$; the analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence $\alpha(n) \to \infty$. As $n \to \infty$, for almost all $m \in [F_n, F_{n+1})$ the distribution of the number of summands in the Zeckendorf decompositions of integers in the subintervals $[m, m + F_{\alpha(n)})$, appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to $1$, $m$ has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and $[0, F_{\alpha(n)})$ to obtain the result, since the summands are known to have Gaussian behavior in the latter interval. % We also prove the same result for more general linear recurrences.Comment: Version 1.0, 8 page

Prevention of secondary stroke in VA: Role of occupational therapists and physical therapists

Occupational therapists (OTs) and physical therapists (PTs) have the opportunity and obligation to advocate secondary stroke prevention via health promotion (HP) behaviors. This prospective survey of Department of Veterans Affairs (VA) OTs and PTs determined whether they know about VA stroke rehabilitation guidelines and whether they integrate secondary stroke prevention into poststroke rehabilitation care. Questions revolved around knowledge of VA guidelines, inclusion of stroke risk-factor modification, and HP education to patients. Thirty-four surveys (45%) were returned from six facilities. Participants included 12 OTs and 22 PTs. Half (53%) of the therapists were aware of the VA guidelines and nearly half (48%) provided HP activities to patients; PTs were significantly more likely to do so than OTs (p = 0.02). Half of the queried therapists were unaware of the VA guidelines; increasing therapists’ education about the guidelines and the necessity of HP and secondary stroke prevention may reduce veterans’ risk of a second stroke. Because many stroke risk factors are modifiable and stroke survivors spend a great deal of time with the rehabilitation therapist, OTs and PTs can and should provide such education to reduce the risk of a second stroke

An effective Chebotarev density theorem for families of fields, with an application to class groups

This talk will present an effective Chebotarev theorem that holds for all but a possible zero-density subfamily of certain families of number fields of fixed degree. For certain families, this work is unconditional, and in other cases it is conditional on the strong Artin conjecture and certain conjectures on counting number fields. As an application, we obtain nontrivial average upper bounds on ℓ-torsion in the class groups of the families of fields

Microgenomic approaches to identify clinically relevant gene signatures that discriminate histologic types of breast carcinomas.

Background: Breast cancer presents itself in a variety of histologic types, and the two most common types are invasive ductal carcinoma (IDC) and invasive lobular carcinoma (ILC). Based on comparative genomic hybridization (CGH) analyses, ILC is more closely related to low grade IDC than it is to intermediate and high grade IDC. Results from the BIG 1-98 trial demonstrate that post-menopausal women who are affected with estrogen receptor positive (ER+) ILC or luminal B (high grade) IDC experience a greater magnitude of benefit when they are treated with the aromatase inhibitor (AI) letrozole compared to treatment with the antiestrogen tamoxifen. To contrast, it has been found in the same trial that women affected with luminal A (low grade) IDC experience more benefit when treated with tamoxifen when compared to letrozole. It is therefore imperative to accurately distinguish low grade IDC from ILC considering their varying responses to adjuvant treatment. Despite genetic evidence suggesting a close relationship between ILC and low grade IDC, a clinically relevant gene set underlying a tumor’s biologic responsiveness to letrozole likely exists. The goal of this study is to use microgenomics to identify a clinically relevant candidate gene set that would discriminate between ILC and low grade IDC rather than relying solely on histomorphology and/or immunohistochemistry for the pathologic diagnosis, especially when conventional tests are conflicting. Methods: Using 247 de-identified human breast carcinoma biopsies collected under standardized, stringent conditions, total RNA was extracted from carcinoma cells procured by laser capture microdissection to perform microarray analyses of approximately 22,000 genes to identify expression signatures associated with breast cancer characteristics. Of the 247 LCM-procured samples, 14 were ER+ ILC, 9 were ER+ low grade IDC, and 43 were ER+ high grade IDC. The other 181 samples were either ER- or of another cancer type other than ILC and IDC. Candidate genes were selected by identifying those that were differentially expressed between ILC and low grade IDC (luminal A) and at the same time, had similar expression levels between ILC and high grade IDC (luminal B). qPCR analyses of whole tissue samples were then utilized to validate the selected gene set. Results: Comparison of microarray data from hormone receptor positive tumors yielded 299 probes that were differentially expressed (p0.01) between ILC and high grade IDC (luminal B). 11 of these 99 genes were initially chosen for further investigation by performing qPCR on whole tissue samples from 21 ILC, 19 low grade IDC and 19 high grade IDC tumors. Our initial analysis revealed expression of the gene coding for heparin-binding EGF like growth factor (HBEGF) and collapsin response mediator protein 1 (CRMP1) may be potential markers for differentiating between ILC and low grade (luminal A) IDC