ABSTRACTS
April 26-27, 2014
Steven J. Miller, Williams College
TITLE: From Sato-Tate distributions to low-lying zeros
ABSTRACT: Physicists developed Random Matrix Theory (RMT) in the 1950sto explain the energy levels of heavy nuclei. A fortuitous meeting over tea atthe Institute in the 1970s revealed that similar answers are found for zeros ofL-functions, and since then RMT has been used to model their behavior. The
distribution of these zeros is intimately connected to many problems in number
theory, from how rapidly the number of primes less than X grows to the classnumber problem to the bias of primes to be congruent to 3 mod 4 and not 1 mod
4. We report on recent progress on understanding the zeros near the central
point, emphasizing the advantages of some new perspectives and models. We
end with a discussion of elliptic curves. We’ll mix theory and experiment andsee some surprisingly results, which lead us to conjecture that a new randommatrix ensemble correctly models the small conductor behavior.

Caroline Turnage-Butterbaugh, University of Mississippi
TITLE: Moments of products of automorphic L-functions
ABSTRACT: Assuming the Generalized Riemann Hypothesis, we will discuss
how to prove upper bounds for moments of arbitrary products of automorphic
L-functions and for Dedekind zeta-functions of Galois number fields on the critical line. As an application, we will use these bounds to estimate the varianceof the coefficients of these zeta- and L-functions in short intervals.

Frank Thorne, University of South Carolina

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TITLE: Zeroes of L-functions outside the critical strip
ABSTRACT: Can an L-function have a zero in its region of absolute convergence? If it is an Euler product, then clearly not. But for a wide classof L-functions not given by Euler products, for example the L-function associated to a modular form which is not a Hecke eigenform, we prove that theseL-functions can and must have zeroes outside the critical strip.
Our work extends work of Saias and Weingartner, who proved this for thedegree 1 case. This is joint work with Andrew Booker.

Dania Zantout, Clemson University
TITLE: A Geometric Notion of Cusp Forms in Higher Genus
Abstract: Let Γ be an arithmetic group. Using the theory of Satake compactification of the Siegel variety Γ \ hn, we define a global operator that projects aSiegel modular form to all the rational boundaries of lower degrees. This gives
us a generalized notion of cusp forms in higher genus that is in parallel to thenotion of elliptic cusp forms.

Nathan McNew, Dartmouth College
TITLE: Some unconventional results in multiplicative combinatorial number
theoryABSTRACT: What is the maximal density of an integer sequence that avoids
geometric progressions? (It’s more than 6/π 2 .) What if we look at the largest
geometric-progression-free subset of the integers up to N ? or modulo N ? How
many multiplications should we expect to make until a product of random
residues modulo N is 0 modulo N ? Is it significantly more than the largest
prime factor? Which primes are we most likely to encounter as this largest
prime factor?

Tom Wright, Wofford College
TITLE: Variants of Korselt’s Criterion2

ABSTRACT: Korselt’s Criterion, a necessary and sufficient condition for Carmichaelnumbers, states that a number n is Carmichael if and only if n is square-free andfor each prime p that divides n, p − 1|n − 1. In this talk, we show that under theassumption of a conjecture about the least prime in an arithmetic progression,
one can prove that for any a ∈ Z, there are infinitely n for which p|n implies
p − a|n − a. This is an improvement of a result of Ekstrom, Pomerance, andThakur.

Michael Mossinghoff, Davidson College,
TITLE: A proof from The Magazine
ABSTRACT: Paul Erdős was well known for labeling a particularly succinct
and insightful justification of a theorem its “proof from The Book”. We presenta problem in number theory, and a solution that may have been drawn from aless ecclesiastical source, which we might dub The Magazine, after the original
definition of the word: a storehouse for a large collection of like things. The
problem concerns the representation of certain algebraic integers (small negative
Pisot and Salem numbers) by very simple polynomials—only 0 and 1 are allowed
as coefficients. We give some indication of the storehouse of cases required totackle the problem, and ask if a Book proof of a more general statement may
still lay hidden. This is joint work with Kevin Hare.

James Diffenderfer, Georgia Southern University
TITLE: Sequences with rational generating functions
ABSTRACT: Let {sn }∞n=0 be a sequence with the rational generating funcP∞(x), with deg(f (x)) < deg(g(x)) and g(x) 6= 0. We presenttion n=0 sn xn = fg(x)a new expression for sn and examine some applications of this expression. In
addition, we present generalized congruences for the sequence sn .

Jesse Thorner, Emory University
TITLE: Bounded Gaps Between Primes in Chebotarev Sets

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ABSTRACT: A new and exciting breakthrough due to Maynard establishes that
there exist infinitely many pairs of distinct primes p1 , p2 with |p1 − p2 | ≤ 600
as a consequence of the Bombieri-Vinogradov Theorem. We apply his general
method to the setting of Chebotarev sets of primes. We study applications ofthese bounded gaps with an emphasis on ranks of prime quadratic twists ofelliptic curves over Q, congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadraticforms.

Brian Sinclair, University of North Carolina at Greensboro
TITLE: Enumeration of p-adic Extensions With a Given Ramification Polygon
Abstract: Let k be a p-adic field. The work of Krasner gives formulae forthe number of extensions of k of a given degree and discriminant. Ramification
polygons are an extension invariant that helps us to study the wild ramificationof an extension. In this talk, we will describe conditions on the ramification
polygon of a totally and wildly ramified extension of k of given degree and discriminant. We will use this to provide a constructive specification of Krasner’s
result to count the number of extensions with a given ramification polygon overk.

Leo Goldmakher, University of Toronto
TITLE: Bounds on the least quadratic nonresidue
ABSTRACT: Attaining strong bounds on the least quadratic nonresidue (modp) is a classical problem, with a history stretching back to Gauss. The approach
which has led to the best results uses character sums, objects which are ubiquitous in analytic number theory. I will discuss character sums, their connection
to the least nonresidue, and some recent work of myself and J. Bober (University of Bristol) on a promising new approach to the problem.

Michael Filaseta, University of South Carolina4

TITLE: 49598666989151226098104244512917
ABSTRACT: A. Cohn showed that if the dr dr−1 . . . d0 is the decimal representation of a prime, then f (x) = dr xr + dr−1 xr−1 + · · · + d0 is irreducible. Let
N be the number in the title. Recently, SamPr Gross and the speaker generalized
Cohn’s result by showing that if f (x) = j=0 dj xj ∈ Z[x] with 0 ≤ dj ≤ N + 1and f (10) prime, then f (x) is irreducible. Further, we showed, by way of anexample, that N + 1 is best possible. In this talk, we focus on recent further related investigations by Morgan Cole, Scott Dunn and the speaker. In particular,
an important property of N will be discussed having connections to factoring
integers with the number field sieve.

Rick Farr, University of North Carolina at Greensboro
TITLE: On Non-integer Stieltjes Constants and Fractional Differentiation
ABSTRACT: γk (a) (k ∈ N ∪ {0}) are the coefficients of the Laurent series expansion of the Hurwitz zeta function ζ(s, a) about the point s = 1. Kreminski
defined these constants for any k ≥ 0 using the notion of fractional differentiation. We discuss fractional differentiation and give a definition of these constants
that is equivalent to Kreminski generalization. From this, we will then prove aconjecture set forth by Kreminski. We will also discuss approximation of theseconstants using Euler-Maclaurin summation.

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