On the distribution of positive and negative values of Hardy's Z-function
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}....
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Published in | Journal of number theory Vol. 174; pp. 189 - 201 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Elsevier Inc
01.05.2017
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Subjects | |
Online Access | Get full text |
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Summary: | 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}. |
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ISSN: | 0022-314X 1096-1658 |
DOI: | 10.1016/j.jnt.2016.10.015 |