1994 result of Smith and Csordas inspires solution to Newman’s Conjecture

Newly (re-)elected department chair Wayne Smith with Varga and our professor emeritus George Csordas in a 1994 paper gave results on the de Bruijn-Newman constant $\Lambda$, where the statement $$\Lambda\le 0$$ is equivalent to the Riemann Hypothesis (some of Riemann’s notes on this shown above). Now in 2018 Terence Tao and Rodgers have shown $$\Lambda\ge 0.$$ Their primary inspiration, judging from the abstract, was Smith et al.’s work:
Wayne Smith in 2012

For each $t \in {\bf R}$, define the entire function $$ H_t(x) := \int_0^\infty e^{tu^2} \Phi(u) \cos(xu)\ du$$ where $\Phi$ is the super-exponentially decaying function $$ \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} – 3\pi n^2 e^{5u} ) \exp(-\pi n^2 e^{4u} ).$$ Newman showed that there exists a finite constant $\Lambda$ (the de Bruijn-Newman constant) such that the zeroes of $H_t$ are all real precisely when $t \geq \Lambda$. The Riemann hypothesis is the equivalent to the assertion $\Lambda \leq 0$, and Newman conjectured the complementary bound $\Lambda \geq 0$.
In this paper we establish Newman’s conjecture. The argument proceeds by assuming for contradiction that $\Lambda < 0$, and then analyzing the dynamics of zeroes of $H_t$ (building on the work of Csordas, Smith, and Varga) to obtain increasingly strong control on the zeroes of $H_t$ in the range $\Lambda< t \leq 0$, until one establishes that the zeroes of $H_0$ are in local equilibrium, in the sense that locally behave (on average) as if they were equally spaced in an arithmetic progression, with gaps staying close to the global average gap size. But this latter claim is inconsistent with the known results about the local distribution of zeroes of the Riemann zeta function, such as the pair correlation estimates of Montgomery.


Mathematics and Motherhood

Two members of the Mathematics Department — new assistant professor Elizabeth Gross and postdoc Piper Harron — have papers in the current issue of the  Journal of Humanistic Mathematics (http://scholarship.claremont.edu/jhm/), a Special Issue on Mathematics and Motherhood .

Dr. Gross wrote “PB&J,”  which explores pregnancy in graduate school, parenting on the tenure-track, division of household labor, and sandwiches.

In “On Contradiction,” Dr. Harron exposes the challenges faced by marginalized parents and the problems inherent in separating our lives into “mathematics” and “everything else” (including the problems inherent in separate journals, special issues like this one, and dedicated panels for the “life” parts).

Strong parallels between the very different pieces emerge, including the incredible time demands of parenthood, the importance of supportive colleagues, and hilarious stories from the trenches (including the photographic reenactment shown from Harron’s piece).

Read Dr. Gross’s essay here: https://tinyurl.com/GrossPBJ.

Read Dr. Harron’s essay here: https://tinyurl.com/HarronContradiction.