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Mean Values of Character Sums

Published online by Cambridge University Press:  20 November 2018

H. L. Montgomery  and
R. C. Vaughan
H. L. Montgomery
Affiliation:
Institute for Advanced Study, Princeton, New Jersey
R. C. Vaughan
Affiliation:
Imperial College, London, England
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For a non-principal Dirichlet character χ modulo q,

Let the Pólya-Vingradov inequality asserts that M(x) < q1/2 log q see [7]. in the opposite direction it is a trivial consequence of lemma 1 below and 1. Parseval's identity that if χ is primitive modulo q, then

We show that on average the latter of these estimates is the more precise.

THEOREM 1. For any real k > 0

where the summation is over all non-principal characters modulo q.

THEOREM 2. For any k > 0,

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 3 , 01 June 1979 , pp. 476 - 487
Copyright
Copyright © Canadian Mathematical Society 1979

References

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