At this year’s Maine-Québec Number Theory Conference, I’m giving a talk on Zeros of half-integral weight Dirichlet series. Here are the slides. I note that the references for the slides are included here at the end.
I’ll also note a few open problems that I don’t know how to handle and that I briefly describe during the talk.
- Is it possible to show that every (symmetrized) Dirichlet series associated to a half-integral weight modular form must have zeros off the critical line? This is true in practice, but seems hard to show.
- Is it possible to determine whether a given Dirichlet series has zeros in the half-plane of absolute convergence? If there is one zero, there are infinitely many – but is there a way of determining if there are any?
- Why does there seem to be a gap around the critical line in zero distribution?
- Can one explain why the pair correlation seems well-behaved (even heuristically)?
If you have any ideas, let me know!