Edmonton: 2-59A, Calgary: BioSci 540B, Vancouver: West Mall Annex 110
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.
