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The Quarterly Journal of Mathematics 2004 55(3):307-324; doi:10.1093/qmath/hah004
© 2004 by Oxford University Press
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A new mean-value result for Dirichlet L-functions and polynomials

Glyn Harman1, Nigel Watt2 and Kam Wong3

1 Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX 2 45 Charles Way, Limekilns, Fife, Scotland KY11 3LH 3 43 Nam Hang Village, Tai Po, New Territories, Hong Kong

Watt's result on upper bounds for the mean value

{int}-TT |Z (1/2+it ) |4 |{sum}m am m-it |2 dt

is generalized to cover

(1)/({phi} (q)) {sum}{chi} (mod q) {int}-TT |L (1/2+it, {chi} ) |4 |{sum}m am {chi} (m) m-it |2 dt,

where {chi} denotes a Dirichlet character (modulo q) and L(s, {chi}) is the corresponding Dirichlet L-function.

Received 30 January 2004.


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