Talks from Perspectives on the Riemann Hypothesis

The Riemann hypothesis is equivalent to the assertion that the entire function $H_0(z) = 1/8 \xi(1+i z/2 )$ has all zeroes on the real line. De Bruijn and Newman studied the deformations $H_t$ of this entire function under the backwards heat equation $\partial_t H_t(z)=-\partial_{zz} H_t(z)$, and showed that there is a real number $\Lambda$, known as the de Bruijn-Newman constant, such that all the zeroes of $H_t$ are real if and only if $t\ge \Lambda$. Thus the Riemann hypothesis is equivalent to the assertion $\Lambda\ge 0$. With Brad Rodgers, we have recently established the complementary bound $\Lambda \ge 0$ (improving upon the previous lower bound of $-1.1 \times 10^{-11}$), and in an ongoing "Polymath" collaboration we are improving the previous upper bound of $1/2$. The former results rely primarily on an analysis of the dynamics of zeroes under heat flow, and the latter on efficient numerical verifications of zero-free regions for the $H_t$; we will present both of these arguments in this talk.

In 1989, Selberg introduced his eponymous class of L-functions, giving rise to a new subfield of analytic number theory in the intervening quarter century. Despite its popularity, a few things have always bugged me about the definition of the Selberg class. I will discuss these nitpicks and describe an attempt at reformulating the Selberg class in terms of distributional identities akin to Weil's "explicit formula". Along the way I will wax philosophical about RH and the converse theorem for automorphic forms.