**Algebra and Number Theory Seminar**

**Emanuel Carneiro**

IMPA, Rio de Janeiro

**Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function**

This talk lies on the interface of analysis and analytic number theory. I will show how to construct a special reproducing kernel Hilbert space related to the Riemann zeta-function and how one can use this space to obtain bounds for the pair correlation of zeros of the zeta-function, extending classical work of P. X. Gallagher (1985). This is a joint work with V. Chandee, M. Milinovich and F. Littmann.

**Tristan Freiberg**

University of Missouri

**Limit points of the sequence of normalized prime gaps**

Let p_n denote the n-th smallest prime number, and let L denote the set of limit points of the sequence of normalized differences between consecutive primes. We show that for k = 50 and for any sequence of k nonnegative real numbers beta_1 < beta_2 < … < beta_k, at least one of the numbers beta_j – beta_i

belongs to L. It follows that more than 2% of all nonnegative real numbers belong to L.

**Ryan Daleida**

Trinity University

**Making imprimitive Dirichlet characters behave primitively**

Given a Dirichlet character \chi mod q, it is traditional to extend \chi to all of Z/qZ by declaring that \chi(n) = 0 when (n,q) \neq 1. When \chi is primitive (i.e. not induced by a Dirichlet character mod d for some proper divisor d of q), this extension endows the associated Gauss sum and L-function with properties that are lost when \chi is imprimitve. In this talk we will introduce a modification to the traditional extension of imprimitive characters which causes them to behave primitively, in the sense that the relevant properties of the Gauss sum and L-function take on the form usually only associated to primitive characters.

**Nathan Jones**

University of Mississippi

**Probability and elliptic curves (I, II and III)**

This expository lecture will be the (first/second/third) of a short series surveying work of Lang and Trotter from the 1970s. For an elliptic curve y^2 = x^3 + ax + b (with a and b integers) and a prime number p, one may consider the elliptic curve modulo p, i.e. one may consider the equation y^2 congruent to x^3 + ax + b modulo p. In particular, it is of wide interest to understand the number N_p of solutions (x,y) modulo p to this congruence, and how this number N_p varies as the prime p varies. In these lectures, we will use probabilistic notions to make very precise conjectures about some aspects of the variation of N_p with p. This talk will be accessible to graduate students.

**Jim Coykendall**

Clemson University

**An Overview of Factorization: Algebraic and Graphical**

Since about 1990, there has been a large amount of effort devoted to the study of factorization in integral domains (as well as in other structures). Much of this study can be interpreted as an attempt to understand how the multiplicative structure of an integral domain “works” when we do not have unique factorization. A classical example is the class group, the size and complexity of which may be interpreted as a measure of “how far” a (Krull) domain is from being a Unique Factorization Domain.

The aim of this talk will be to give an overview of recent factorization theory. We will highlight some basic definitions, examples, and results. We will also highlight some more recent results that lend themselves to visualizations and have interesting connections to graph theory.

**Tsz Ho Chan**

University of Memphis

**A Look at the Modular Hyperbola** (pdf)

**David Farmer**

American Institute of Mathematics

**Finding and calculating L-functions** (pdf)

**Stephan Baier**

University of Bristol

**Subconvexity bounds for L-functions** (pdf)

** Micah Milinovich**

University of Mississippi

**The Riemann Hypothesis…** (pdf)

**Tsz Ho Chan**

University of Memphis

**Sums of two Squares and Almost Squares** (pdf)

**Ryan Daileda**

Trinity University

**Maximal Class Numbers of CM Number Fields** (pdf)

**Andrew Odlyzko**

University of Minnesota

**Zeros of the Riemann zeta function: computations and implications** (pdf)

** Hung Manh Bui**

Oxford University

**Gaps between consecutive zeros of the Riemann zeta-function** (pdf)

**Zhu Cao**

University of Mississippi

**Integer Matrix Exact Covering Systems and Product Identities for Theta Functions** (pdf)

**Neil Epstein**

University of Michigan

**Closure Operations on Ideals in Commutative Rings** (pdf)

**Zhu Cao**

University of Mississippi

**A new proof of Winquist’s Identity** (pdf)

**Zhu Cao**

University of Mississippi

**A Proof of Lagrange’s Four-Square Theorem** (pdf)

**Micah Milinovich**

University of Mississippi

**Distribution of the Prime Numbers** (pdf)