Number Theory Seminar: Linear (in)dependence of zeros of L-functions

In this hastily prepared talk, I will describe some preliminary results of Nathan Ng and myself that concern linear dependencies (with integer coefficients) among zeros of Dirichlet L-functions. We can show, for example, that given a Dirichlet L-function and an arithmetic progression of points on the critical line Re(s) = 1/2, a large number of points in the arithmetic progression are not zeros of the L-function. Furthermore, given a fixed linear form F in n variables, we show (assuming the Riemann hypothesis) that a large number of points of the form 1/2 + iF(gamma_1, ..., gamma_n) are not zeros of the Riemann zeta function, where the gamma_j are imaginary parts of such zeros. We also describe a theorem about prime number races that is linked to linear (in)dependence of zeros of Dirichlet L-functions.