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

• Published in: Journal of number theory Vol. 174; pp. 189 - 201
• Main Authors: Gonek, Steven M., Ivić, Aleksandar
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.05.2017
• Subjects: Distribution of positive and negative values; Hardy's function; Riemann zeta-function; 11M06; Distribution of positive and negative values; Hardy's function; Riemann zeta-function
• ISSN: 0022-314X; 1096-1658
• DOI: 10.1016/j.jnt.2016.10.015

We investigate the distribution of positive and negative values of Hardy's functionZ(t):=ζ(12+it)χ(12+it)−1/2,ζ(s)=χ(s)ζ(1−s). In particular we prove thatμ(I+(T,T))≫Tandμ(I−(T,T))≫T, where μ(⋅) denotes Lebesgue measure andI+(T,H)={T<t⩽T+H:Z(t)>0},I−(T,H)={T<t⩽T+H:Z(t)<0}.