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

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

On a conjecture for ℓ\ell-torsion in class groups of number fields: from the perspective of moments

It is conjectured that within the class group of any number field, for every integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the full strength of this conjecture remains open, and even partial progress is limited. Significant recent progress toward average versions of the $\ell$-torsion conjecture has crucially relied on counts for number fields, raising interest in how these two types of question relate. In this paper we make explicit the quantitative relationships between the $\ell$-torsion conjecture and other well-known conjectures: the Cohen-Lenstra heuristics, counts for number fields of fixed discriminant, counts for number fields of bounded discriminant (or related invariants), and counts for elliptic curves with fixed conductor. All of these considerations reinforce that we expect the $\ell$-torsion conjecture is true, despite limited progress toward it. Our perspective focuses on the relation between pointwise bounds, averages, and higher moments, and demonstrates the broad utility of the "method of moments.

Data Sonification from the Desktop: Should Sound Be Part of Standard Data Analysis Software?

The design of auditory formats for data display is presently focused on applications for blind or visually impaired users, specialized displays for use when visual attention must be devoted to other tasks, and some innovative work in revealing properties of complex data that may not be effectively rendered by traditional visual means. With the availability of high-quality and flexible sound production hardware in standard desktop computers, the potential exists for using sound to represent characteristics of typical “small and simple” samples of data in routine data inspection and analysis. Our research has shown that basic properties of simple functions, distribution properties of data samples, and patterns of covariation between two variables can be effectively displayed by simple auditory graphs involving patterns of pitch variation over time. While such developments have implications for specialized applications and populations of users, these displays are easily comprehended by normal users with minimal practice. Providing further software enhancement to encourage exploration of data representation by sound may lead to a variety of useful creative developments in data display technology

Market survey results for alternate sensor communications

This document presents the results of a system analysis and market survey of commercially available alarm communication systems for potential use as an alternate sensor communication system. Only those systems that report alarm/sensor information to a central control panel were considered. The communication systems surveyed include wireless radio frequency (RF) systems, spread spectrum systems, fiber optic systems, twisted pair/copper wire, cellular systems, and other types of communication equipment. All systems are commercially available, and most information was obtained by telephone conversations with the manufacturer, personal interviews at security conferences, and countless reviews of the manufacturers` data sheets. Many systems were identified, but only those that met a minimum set of system requirements were included. Other systems that appeared to be applicable usually did not provide adequate data encryption or could not interface directly to the system. While such features could be incorporated using additional hardware, doing so would make the system more expensive and conflict with the idea of purchasing a single unit that meets the minimum set of requirements. Several systems greatly exceed the scope of this project and utilizing such systems would mean investing in more capacity than is really needed

Hilbert transforms and the equidistribution of zeros of polynomials

We improve the current bounds for an inequality of Erdős and Turán from 1950 related to the discrepancy of angular equidistribution of the zeros of a given polynomial. Building upon a recent work of Soundararajan, we establish a novel connection between this inequality and an extremal problem in Fourier analysis involving the maxima of Hilbert transforms, for which we provide a complete solution. Prior to Soundararajan (2019), refinements of the discrepancy inequality of Erdős and Turán had been obtained by Ganelius (1954) and Mignotte (1992)

The Ursinus Weekly, March 6, 1969

Officers inducted; Emig emphasizes new responsibility • American dream, brotherhood highlight Festival of the Arts • USGA officers interviewed; Communication gap stressed • Placement interviews scheduled • 200 attend Lorelei; Fischer crowned king • Semi-formal ball highlights junior-senior weekend • Editorial: Potpourri - Or, Could this ever happen here? • State set to punish protestors; Laws threaten civil liberties • Letters to the editor • Remark • Lantern literary lapse termed titanic bomb • Staying out of uniform: A practical guide for the Ursinus male, part I • Whose risk? • Columnist proposes solution to UC\u27s dearth of black students • Albright draws Fifth Dimension • Racism symposium set for March 20 with Lincoln Univ. • Dorm lounges opened; Slacks rule slackened • Rice discusses status for Phi Beta Kappa • Ursinus grapplers rally to win over Drexel, 22-13 • New gym complex to include pool • Carson paces Bearettes over Gettysburg • Badminton team wins six in row • Trackmen run at Swarthmore and Delaware • Gillespie hits 37 points to lead Bears over Swarthmore in finale • Girl swimmers win over Penn and Elizabethtown • Junior varsity hoopmen finish with 11-6 record • Dickinson frat system defended • UC celebrates centennial with unique innovation • Forum features Howellhttps://digitalcommons.ursinus.edu/weekly/1171/thumbnail.jp