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Bounded Gaps Between Primes in Chebotarev Sets
A new and exciting breakthrough due to Maynard establishes that there exist
infinitely many pairs of distinct primes with as
a consequence of the Bombieri-Vinogradov Theorem. In this paper, we apply his
general method to the setting of Chebotarev sets of primes. We study
applications of these bounded gaps with an emphasis on ranks of prime quadratic
twists of elliptic curves over , congruence properties of the
Fourier coefficients of normalized Hecke eigenforms, and representations of
primes by binary quadratic forms.Comment: 15 pages. Referee comments implemented. Research in the Mathematical
Sciences 2014, 1:
Levels of Distribution and the Affine Sieve
This article is an expanded version of the author's lecture in the Basic
Notions Seminar at Harvard, September 2013. Our goal is a brief and
introductory exposition of aspects of two topics in sieve theory which have
received attention recently: (1) the spectacular work of Yitang Zhang, under
the title "Level of Distribution," and (2) the so-called "Affine Sieve,"
introduced by Bourgain-Gamburd-Sarnak.Comment: 34 pages, 2 figure
Expanding graphs, Ramanujan graphs, and 1-factor perturbations
We construct (k+-1)-regular graphs which provide sequences of expanders by
adding or substracting appropriate 1-factors from given sequences of k-regular
graphs. We compute numerical examples in a few cases for which the given
sequences are from the work of Lubotzky, Phillips, and Sarnak (with k-1 the
order of a finite field). If k+1 = 7, our construction results in a sequence of
7-regular expanders with all spectral gaps at least about 1.52
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