Barry Smith
Associate Professor of Mathematical Sciences
Divisibility properties of the values of partial zeta functions at non-positive integers, Mathematische Zeitschrift, Volume 270, April 2012, Pages 645-657 — a study of a modification of a conjecture of David Hayes concerning the exact denominators of values of partial zeta functions at non-positive integers.
Non-cyclic class groups and the Brumer-Stark conjecture, Journal of Number Theory, Volume 132, Issue 2, February 2012, Pages 348-370 — extends work of Greither-Roblot-Tangedal to prove new cases of the Brumer-Stark conjecture for extensions of number fields of degree 2p.
Reducing quadratic forms by kneading sequences, Journal of Integer Sequences, Volume 17, December 2014, Article 14.11.8 — provides a one-to-one correspondence between Zagier-reduced binary quadratic forms of fixed nonsquare discriminant D>0 and certain sequences of positive integers. In (6) below, this construction is modified to show that it is essentially giving minimal periods of certain continued fractions, and the conjectures in this paper are then proved.
End-symmetric continued fractions and quadratic congruences, Acta Arithmetica, Volume 167, February 2015, Pages 173-187 — shows that for a fixed integer n (not equal to 2 or -2), the congruence x^2 + nx + 1 mod a has the solution b with 0 < b < a if and only if the rational number a/b has continued fraction expansion with sequence of quotients having one of a finite number of asymmetry types.
(with C. Doran and S. Lu) New algorithms for modular inversion and representation by the form x^2 + 3xy + y^2, Involve: a Journal of Mathematics, Volume 10, Issue 4, April 2017, pages 541-554 — reveals patterns in quotients and remainders of the Euclidean algorithm with special inputs, filling in some special cases not covered by (3); gives algorithms to compute modular inverses and, for positive integers representable by the binary quadratic form x^2+3xy+y^2, for computing the unique such representation by positive integers x and y.
Constructing minimal periods of quadratic irrationalities in Zagier’s reduction theory, Journal of Number Theory, Volume 187, June 2018, pages 1-26 — constructs the minimal periods of Denjoy continued fractions (all quotients are 0 or 1) of certain quadratic irrationalities from corresponding binary quadratic forms that are reduced in Zagier’s sense. Also shows how to embed Gauss’s reduction theory into Zagier’s.
Divisibility of partial zeta values at zero for degree 2p extensions, submitted — an extension of the results in (1) to the smallest extensions with p-power roots of unity having nontrivial cohomology.