On the Brun-Titchmarsh Theorem

The Brun-Titchmarsh theorem shows that the number of primes $\le x$ which are congruent to $a\pmod{q}$ is $\le (C+o(1))x/(\phi(q)\log{x})$ for some value $C$ depending on $\log{x}/\log{q}$. Different authors have provided different estimates for $C$ in different ranges for $\log{x}/\log{q}$, all of which give $C>2$. We show that one can take C=2 provided that $\log{x}/\log{q}\ge 8$. Without excluding the possibility of an exceptional Siegel zero, we cannot have $C<2$ and so this result is best-possible in this sense. We obtain this result using analytic methods developed in the study of Linnik's constant. In particular, we obtain explicit bounds on the number of zeroes of Dirichlet $L$-functions with real part close to 1 and imaginary part of size O(1).Comment: 47 Page

Ethics and health care 'underfunding'

There are continual “crises” in health care systems worldwide as producer and patient groups unify and decry the “underfunding” of health care. Sometimes this cacophony is the self interest of profit seeking producers and often it is advocacy of unproven therapies. Such pressure is to be expected and needs careful management by explicit rationing criteria which determine who gets access to what health care. Science and rationality, however, are unfortunately, rarely the rules of conduct in the medical market-place

Community Reclamation: the Hybrid Building

Reclamation of a city involves reusing abandoned buildings in conjunction with new construction. These negative spaces of disuse generated by a changing infrastructure are often overlooked or destroyed. If they are instead viewed as positive spaces for reuse, a city’s infrastructure and its residents can adapt and grow. Recognizing these newly positive spaces produces a chance to examine what social needs of the community are not being met. Pushing the modern concept of the hybrid building creates a unique opportunity; flexibility of use derived from flexibility of space. A community building can best serve the social needs of its residents by having the ability to adapt to changes in those needs

Primes with restricted digits

Let $a_0\in\{0,\dots,9\}$. We show there are infinitely many prime numbers which do not have the digit $a_0$ in their decimal expansion. The proof is an application of the Hardy-Littlewood circle method to a binary problem, and rests on obtaining suitable `Type I' and `Type II' arithmetic information for use in Harman's sieve to control the minor arcs. This is obtained by decorrelating Diophantine conditions which dictate when the Fourier transform of the primes is large from digital conditions which dictate when the Fourier transform of numbers with restricted digits is large. These estimates rely on a combination of the geometry of numbers, the large sieve and moment estimates obtained by comparison with a Markov process.Comment: 70 page

Anthropology of the Crowd, Blog 8

Student blog posts from the Great VCU Bike Race Book

Just What You’re Looking For

Overview: Death is inevitable. It’s something that we are all going to experience at some point in our lives, and it’s all just a matter of when. Many people find death to be too abstract and frightening to contemplate, so it becomes an idea that is displaced to back of their minds to deal with later. For some people that later time comes before they know it. It comes before they are able to grasp the idea of what death is and therefore cannot understand it. Or it can come unexpectedly and without planning. Feeling alone, sad, angry, and miserable, the person left is without any ways to deal with the incident. That person could be twelve years old. According to the website, Grief Watch, “ …around one in ten adolescents between the ages of ten and eighteen [have] experienced the loss of a close loved one” (n.p.). These children and young adults may have lost a grandparent, parent, aunt, uncle, or a friend. How do children specifically deal with a loss? One tool is young adult literature, also known as YAL. YAL is geared towards adolescents and intended to relate to their young lives. It includes many obstacles that young adults are facing today such as death. One young adult novel that teens can relate to is, The Absolutely True Diary of a Part Time Indian by Sherman Alexie. The main character, Junior, experiences several challenges of living on a reservation while going to a nearby white public school. He overcomes the death of his honorable grandmother, close family friend, and beloved sister. In another novel, Looking for Alaska by John Green, a group of students at a boarding school look for answers for the death of a close friend. Both novels include relatable characters that adolescents can look to for ways to cope with their own loss. By looking at these two novels, we can see that coping with death is a challenge for adolescents, which most people don’t see; this is important because adolescents can use young adult literature to grieve in a positive way. Young adult literature is written for the interest of adolescents by relating to their lives through the characters, their hardships, interests, and culture. Dr. Jonathon Ostenson, a proponent of YAL claims, “Young adult literature is most succinctly defined by Bucher and Hinton (2010) as a work of any genre that, ‘provides a unique adolescent point of view, and reflects the concerns, interests, and challenges of young adults’” (n.p.). Thus, YAL is written to reflect the lives of young adults. Accordingly, when children can relate to what they are reading, then the novel and story becomes of interest to them. This is imperative for teens who dislike reading. Yet, when they read YAL, it becomes a tool that can be used for development. In Lorna Collier’s article, “YA Literature-Where Teens Find Themselves” she writes, “ YA lit is an invaluable resource in today’s English classrooms, engaging students with relevant topics, relatable characters, and accessible language” (6). This is because young adult literature is a relatively new category of literature that reflects the lives of children today more than the classics that are required of many students to read. Specifically, it is more suitable in their lives than reading about challenges that young adults faced in a time that is irrelevant to the 21st century. Thus, adolescents can learn as much, if not more, from YAL than other resources available to them

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