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)?
Info on how to comment
To make a comment, please send an email using the button below. Your email address won't be shared (unless you include it in the body of your comment). If you don't want your real name to be used next to your comment, please specify the name you would like to use. If you want your name to link to a particular url, include that as well.
bold, italics, and plain text are allowed in comments. A reasonable subset of markdown is supported, including lists, links, and fenced code blocks. In addition, math can be formatted using
$(inline math)$or$$(your display equation)$$.Please use plaintext email when commenting. See Plaintext Email and Comments on this site for more. Note also that comments are expected to be open, considerate, and respectful.