Pell conics and quadratic reciprocity

We give a proof of quadratic reciprocity, based on the arithmetic of conics. The proof works in all cases, and the calculations are remarkably simple

Quadratic Residues and Non-Residues: Selected Topics

Number theory as a coherent mathematical subject started with the work of Fermat in the decade from 1630 to 1640, but modern number theory, that is, the systematic and mathematically rigorous development of the subject from fundamental properties of the integers, began in 1801 with the appearance of the landmark text of Gauss, Disquisitiones Arithmeticae. A major part of the Disquisitiones deals with quadratic residues and nonresidues. Beginning with these fundamental contributions of Gauss, the study of quadratic residues and nonresidues has subsequently led directly to many of the key ideas and techniques that are used everywhere in number theory today, and the primary goal of these lectures is to use this study as a window through which to view the development of some of those ideas and techniques. In pursuit of that goal, we will employ methods from elementary, analytic, and combinatorial number theory, as well as methods from the theory of algebraic numbers.Comment: xi+265 pp., 4 tables, 20 figures in Lecture Notes in Mathematics no. 2171, Springer, New York, 201

On \u3ci\u3e L\u3c/i\u3e-functions and the 1-Level Density

We begin with the classical study of the Riemann zeta function and Dirichlet L-functions. This includes a full exposition on one of the most useful ways of exploiting their connection with primes, namely, explicit formulae. We then proceed to introduce statistics of low-lying zeros of Dirichlet L-functions, discussing prior results of Fiorilli and Miller (2015) on the 1-level density of Dirichlet L-functions and their achievement in surpassing the prediction of the powerful Ratios Conjecture. Finally, we present our original work partially generalizing these results to the case of Hecke L-functions over imaginary quadratic fields

Real equiangular lines and related codes

We consider real equiangular lines and related codes. The driving question is to find the maximum number of equiangular lines in a given dimension. In the real case, this is controlled by combinatorial phenomena, and until only very recently, the exact number has been unknown. The complex case appears to be driven by other phenomena, and configurations are conjectured always to meet the absolute bound of d^2 lines in dimension d. We consider a variety of the techniques that have been used to approach the problem, both for constructing large sets of equiangular lines, and for finding tighter upper bounds. Many of the best-known upper bounds for codes are instances of a general linear programming bound, which we discuss in detail. At various points throughout the thesis, we note applications in quantum information theory