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

Online Catalogs and User Education

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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

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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

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Flame detector operable in presence of proton radiation

A detector of ultraviolet radiation for operation in a space vehicle which orbits through high intensity radiation areas is described. Two identical ultraviolet sensor tubes are mounted within a shield which limits to acceptable levels the amount of proton radiation reaching the sensor tubes. The shield has an opening which permits ultraviolet radiation to reach one of the sensing tubes. The shield keeps ultraviolet radiation from reaching the other sensor tube, designated the reference tube. The circuitry of the detector subtracts the output of the reference tube from the output of the sensing tube, and any portion of the output of the sensing tube which is due to proton radiation is offset by the output of the reference tube. A delay circuit in the detector prevents false alarms by keeping statistical variations in the proton radiation sensed by the two sensor tubes from developing an output signal

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

A device for the objective assessment of ADHD using eye movements

Attention deficit hyperactivity disorder (ADHD) is a commonly diagnosed psychiatric disorder characterized by lack of focus, self-control,and hyperactivity. ADHD is difficult to diagnose without extensive observation by an expert, and even then is often misdiagnosed. Current methods of pediatric diagnosis rely on subjective measures of activity and behavior relative to other children [3]. Proper diagnosis is critical in preventing unnecessary prescription of the powerful, habit-forming nature of the drugs used to manage ADHD, such as Adderall and Ritalin [1][5]. Research has shown that patients with ADHD show abnormalities in reading tests and antisaccade tests, as these tests gauge ability to focus and suppress impulsive behavior [2][6][4]. This project proposes to create a dedicated device that will use eye movement analysis to accurately and objectively screen children for ADHD. The device will be inexpensive and easy to use for school nurses, optometrists, and primary care physicians. First, research was conducted to decide the type of eye tracker to build, the tests that would be run, the layout of the device, and the type of headgear to use. After the preliminary research was completed, it was decided that a limbus eye tracker would best fit the needed functionality of the device. Limbus tracking is both more accurate in horizontal tracking and less costly than other systems. A basic circuit diagram has been created and circuit parts have been ordered. The IR LED and phototransistors have been tested and appear to be working properly, but further testing will be conducted and mounting for the components will be constructed. One problem encountered was the selection of a computational module that incorporates our needs for digital I/O, A/D conversion, significant processing power and speed, DOS-basedoperating system, and VGA output. No single board computer yet found incorporates all these features in one module without being too costly. The team is awaiting a decision concerning Sternheimer funding before exploring the use of more cost-effective strategies. Another point of discussion among the team was how to affix the device to a child’s head or keep a child’s head still enough for the eye tracker to be accurate. The result was a preliminary design utilizing safety glasses. The next steps in this project include deciding upon a single board computer and ordering it and ordering more circuit parts and safety glasses. While these parts come in, the circuit design can be enhanced, an approach for the programming portion will be created.https://scholarscompass.vcu.edu/capstone/1007/thumbnail.jp

A new approach to solving the multimode kinetics equations

Cover reads: By Joe C. Turner [sic], Allan F. HenryAlso issued as a Ph. D. thesis in the Department of Nuclear Engineering, 1972Includes bibliographical references (leaves 95-97)AEC AT(11-1)--305