A Pythagorean Introduction to Number Theory : Right Triangles, Sums of Squares, and Arithmetic

In the ?rst section of this opening chapter we review two different proofs of the PythagoreanTheorem,oneduetoEuclidandtheotheroneduetoaformerpresident oftheUnitedStates,JamesGar?eld.Inthesamesectionwealsoreviewsomehigher dimensional analogues of the Pythagorean Theorem. Later in the chapter we de?ne Pythagorean triples; explain what it means for a Pythagorean triple to be primitive; and clarify the relationship between Pythagorean triples and points with rational coordinates on the unit circle. At the end we list the problems that we will be interested in studying in the book. In the notes at the end of the chapter we talk about Pythagoreans and their, sometimes strange, beliefs. We will also brie?y review the history of Pythagorean triples

Multiple Mixing for adele groups and rational points

We prove an asymptotic formula for the number of rational points of bounded height on projective equivariant compactifications of H\G, where $H$ is a connected simple algebraic group embedded diagonally into $G := H^n$

On certain multiple Dirichlet series

In this paper we study the analytic properties of a multiple Dirichlet series associated to the prehomogeneous vector space of binary cubic forms.Comment: 21 page