The density of integers of the form u^2 + nv^2

Speaker: Dr. David Brink (UCD)

Time: 4:00PM
Date: Wed 3rd February 2010

Location: Mathematical Sciences Seminar Room

Let B(x) denote the number of positive integers up to x of the form u^2+v^2. It is a famous result of Landau that B(x) grows asymptotically as b x/sqrt{log x} with the constant b=0.7642.... In his doctoral thesis, Landau's student Bernays showed more generally that the number B_n(x) of positive integers up to x of the form u^2+nv^2 grows as b_n x/sqrt{log x} with some constant b_n. In this talk we discuss a conjecture of Shanks and Schmid that b_2=0.8728... is maximal among these constants.

(This talk is part of the K-Theory, Quadratic Forms and Number Theory series.)