Do the zeros of one $L$-function know about the others?MPIM
by
MPIM, Vivatsgasse, 7 - Lecture Hall
Max Planck Institute for Mathematics
Number theory lunch seminar
In principle, the zeros of two distinct Dirichlet $L$-functions ought
to be unrelated. In fact, they turn out to be far more entangled than one expects, though not always. My talk will survey some recent results in this circle of ideas, including Linnik--Sprind\v{z}uk-type equivalences that reduce GRH to the vertical distribution of the zeros of a single $L$-function. I will also describe in greater depth some joint work with Kyle Loftus evaluating the twisted moment $\sum_{\rho} x^{\rho} L(\rho,\chi_1)$ over the zeros $\rho$ of a second $L$-function $L(s,\chi_2)$. Our results are unconditional and hold in short windows, and they imply that no nontrivial linear combination of Dirichlet $L$-functions vanishes on the entire zero set of another.