On the distribution of positive and negative values of Hardy's $Z$-function

TL;DR

This paper studies the distribution of positive and negative values of Hardy's Z-function, proving that both occur with Lebesgue measure proportional to T within certain intervals, indicating frequent sign changes.

Contribution

It establishes lower bounds on the measure of intervals where Hardy's Z-function is positive or negative, advancing understanding of its oscillatory behavior.

Findings

Both positive and negative values of Z(t) occur with measure proportional to T.

The results imply frequent sign changes of Z(t) over large intervals.

The paper provides quantitative bounds on the distribution of Z(t)'s signs.

Abstract

We investigate the distribution of positive and negative values of Hardy's function $$ Z(t) := \zeta(1/2+it){\chi(1/2+it)}^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s). $$ In particular we prove that $$ \mu\bigl(I_{+}(T,T)\bigr) \;\gg T\; \qquad \hbox{and}\qquad \mu\bigl(I_{-}(T, T)\bigr) \; \gg \; T, $$ where $\mu(\cdot)$ denotes the Lebesgue measure and \begin{align*} { I}_+(T,H) &\;=\; \bigl\{T< t\le T+H\,:\, Z(t)>0\bigr\}, { I}_-(T,H) &\;=\; \bigl\{T< t\le T+H\,:\, Z(t)<0\bigr\}. \end{align*}