## Newman's Conjecture in Various Settings

with A. Chang and S. J. Miller

Journal of Number Theory, Vol. 144, (2014) pp. 70-91.

Journal of Number Theory, Vol. 144, (2014) pp. 70-91.

**Abstract:**De Bruijn and Newman introduced a deformation of the Riemann zeta function $\zeta(s)$, and found a real constant $\Lambda$ which encodes the movement of the zeros of the $\zeta(s)$ under the deformation. The Riemann hypothesis is equivalent to $\Lambda\leq0$. Newman conjectured that $\Lambda\geq0$, remarking "the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so". Previous work could only handle $\zeta(s)$ and quadratic Dirichlet $L$-functions, obtaining lower bounds very close to zero ($-1.14541\cdot10^{-11}$ for $\zeta(s)$ and $-1.17\cdot10^{-7}$ for quadratic Dirichlet $L$-functions). We generalize to automorphic $L$-functions and function field $L$-functions, and explore the limits of these techniques. If $\mathcal{D}\in\mathbb{Z}[T]$ is a square-free polynomial of degree $3$ and $D_{p}$ the polynomial in $\mathbb{F}_{p}[T]$ obtained by reducing $\mathcal{D}$ modulo $p$, we prove the Newman constant $\Lambda_{D_{p}}$, equals $\log\tfrac{|a_{p}(\mathcal{D})|}{2\sqrt{p}}$; by Sato-Tate (if the curve is non-CM) there exists a sequence of primes such that $\lim_{n\rightarrow\infty}\Lambda_{D_{p_{n}}}=0$. We end by discussing connections with random matrix theory.