| Abstract | In this talk, I will discuss recent refinements and explicit versions of these results, in joint work with Griffin, Ono, Thorner, Tripp, and Wagner. I will then discuss other types of "Riemann Hypotheses" connected to the world of modular forms, in particular the Riemann Hypothesis for Period Polynomials (RHPP) due to Jin, Ma, Ono, and Soundararajan. I will discuss interpretations of this which were first speculated by Manin, in joint work with Ono and Sprung, and I will outline recent generalizations to higher L-derivatives and Hilbert modular forms in work due to myself and Diamantis and Babei and Wagner. We will finish with a discussion of the general picture and which cases are currently proven and which remain conjectural. |