On the distribution of values of the argument of the Riemann zeta-function
Let S(t):=1πargζ(12+it). We prove that, for T27/82+ε⩽H⩽T, we havemes{t∈[T,T+H]:S(t)>0}=H2+O(Hlog3Tεlog2T), where the O-constant is absolute. A similar formula holds for the measure of the set with S(t)<0, where logkT=log(logk−1T). This result is derived from an asymptotic formula for the...
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Published in | Journal of number theory Vol. 200; pp. 96 - 131 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Elsevier Inc
01.07.2019
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Subjects | |
Online Access | Get full text |
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Summary: | Let S(t):=1πargζ(12+it). We prove that, for T27/82+ε⩽H⩽T, we havemes{t∈[T,T+H]:S(t)>0}=H2+O(Hlog3Tεlog2T), where the O-constant is absolute. A similar formula holds for the measure of the set with S(t)<0, where logkT=log(logk−1T). This result is derived from an asymptotic formula for the distribution of values of S(t), which is uniform in the relevant parameters, and this is of crucial importance. This in fact depends on the distribution of values of the Dirichlet polynomial which approximates S(t), namely (p denotes primes)Vy(t)=∑p⩽ysin(tlogp)p. |
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ISSN: | 0022-314X 1096-1658 |
DOI: | 10.1016/j.jnt.2018.12.011 |