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Hybrid bounds for Dirichlet's L-function

Published online by Cambridge University Press:  17 January 2001

M. N. HUXLEY  and
N. WATT
M. N. HUXLEY
Affiliation:
School of Mathematics, University of Wales College of Cardiff, Senghenydd Road, Cardiff CF2 4AG
N. WATT
Affiliation:
45 Charles Way, Limekilns, Fife KY11 3LH
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Abstract

This is a paper about upper bounds for Dirichlet's L-function, L(s, χ), on its critical line (s + s¯ = 1). It is to be assumed throughout that, unless otherwise stated, the Dirichlet character, χ, is periodic modulo a prime, r, and is not the principal character mod r. Our main theorem below shows that, if ε > 0, then

formula here

(where A is an absolute constant), for 0 < α = (log r)/(log t) [les ] 2/753 − ε. Somewhat weaker bounds are obtained for other cases where 0 < α [les ] 11/180 − ε. Note that in [13] it was shown that, for 0 < α [les ] 2/57,

formula here

Our main theorem is a corollary of the new bounds we prove for certain exponential sums, S, with a Dirichlet character factor:

formula here

where M2 [les ] 2M and f(x) is a real function whose derivatives satisfy certain conditions restricting their size.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 129 , Issue 3 , November 2000 , pp. 385 - 415
Copyright
© 2000 Cambridge Philosophical Society

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