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The asymptotic behaviour of the mean-square of fractional part sums

Published online by Cambridge University Press:  20 January 2009

Manfred Kühleitner  and
Werner Georg Nowak
Manfred Kühleitner
Affiliation:
Institut für Mathematik u. Ang. Stat., Universität für Bodenkultur, A-1180 Wien, Austria (kleitner@mail.boku.ac.at; nowak@mail.boku.ac.at)
Werner Georg Nowak
Affiliation:
Institut für Mathematik u. Ang. Stat., Universität für Bodenkultur, A-1180 Wien, Austria (kleitner@mail.boku.ac.at; nowak@mail.boku.ac.at)
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Abstract

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In this article we consider sums S(t) = Σnψ (tf(n/t)), where ψ denotes, essentially, the fractional part minus ½ f is a C4-function with f″ non-vanishing, and summation is extended over an interval of order t. For the mean-square of S(t), an asymptotic formula is established. If f is algebraic this can be sharpened by the indication of an error term.

Keywords

fractional partsmean squarelattice pointsexponential sums
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 2 , June 2000 , pp. 309 - 323
Copyright
Copyright © Edinburgh Mathematical Society 2000

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