Smooth numbers in short intervals

Assuming the Riemann hypothesis we demonstrate the existence of smooth numbers in certain short intervals.Comment: 6 page

Nonvanishing of quadratic Dirichlet L-functions at s=1/2

We show that for a positive proportion of fundamental discriminants d, L(1/2,chi_d) != 0. Here chi_d is the primitive quadratic Dirichlet character of conductor d.Comment: 42 pages, published versio