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fractional parts

Proceedings of the Edinburgh Mathematical Society (1)

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The asymptotic behaviour of the mean-square of fractional part sums
Manfred Kühleitner, Werner Georg Nowak
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 43 / Issue 2 / June 2000

Published online by Cambridge University Press:

20 January 2009, pp. 309-323
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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.