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Journal of the London Mathematical Society 1976 s2-13(1):53-56; doi:10.1112/jlms/s2-13.1.53
© 1976 by London Mathematical Society
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© Oxford University Press

An Imperfect Hybrid Zero-Density Theorem

M. N. Huxley

University College Cardiff, Wales, Great Britain CF1 1XL

Let Q, T≥1,{alpha}≥1/2. The number of zeros ß+iy in the region ß≥{alpha}|y|≤T of all Dirichlet L-functions formed with primitive characters to moduli ≤ Q is

O((QT)40(1–{alpha})/9+{varepsilon})for any {varepsilon} > 0. The imperfection lies in the large exponent of T; the earlier estimate

O((Q2T)12(1–{alpha})/5+{varepsilon})

is not superseded if T is large. The proof uses Montgomery's theory of finite Dirichlet series and a function-theoretic lemma of Littlewood.


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