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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.


Open Educational Resources for Statistics and Data Science

OER at UH Manoa is growing and the Mathematics is participating with free online (and cheap hard copy) textbooks for MATH 111-112 and MATH 372 on the books and in the works.

The MATH 372 textbook, tentatively entitled Statistics and Hidden Markov Models for Calculus Students, is authored by Professor Bjørn Kjos-Hanssen and PhD student Samuel Birns.

The book is being developed during Fall 2018. It is a supplement to OpenIntro Statistics (3rd Edition), a free textbook which describes calculus in a mathematics-rich but calculus-free way. Reading it, one will encounter many practical examples and probability calculations, but may never learn that the standard normal probability density function is
MATH 372 is our junior-level elementary probability and statistics course and is intended for students who have already taken Calculus I and II. (Students who have taken Calculus III and IV can learn why the normal distribution has that $\frac1{\sqrt{2\pi}}$ factor in MATH 471 and MATH 472, our senior-level courses on the subject.)

A hidden message in the book title is that linear algebra is not a required prerequisite. Thus, we do not go into the typical machine learning material. However, the book includes a chapter on Hidden Markov Models and an associated complexity notion studied in Kjos-Hanssen’s recent Experimental Mathematics article.

MATH 372 is projected to become a high-enrollment course starting in Spring 2019 when Information & Computer Sciences starts to require it for their data science concentration.

Table of Contents:

  1. Data
  2. Probability
  3. Distributions
  4. Inference
  5. Inference for numerical data
  6. Inference for categorical data
  7. Linear regression
  8. Model selection

The textbook projects are funded by Outreach College as one of their OER Grant Projects.