In 1961, answering a problem proposed by N.J.Fine, Besicovitch constructed an example of a non trivial real continous function f on [0,1] which is odd with respect to the point 1/2 and with the property that the sum of the values of the function in the points a/n, with a=1,...,n is equal to 0 for every natural n. Bateman and Chowla in 1963 pointed out that more explicit function (trigonometric cosine series with coefficients depending on Liouville's function and Moebius function) share the same property.In this paper we show that a class of functions arising as formal limits of a certain finite minimizing problem also have this strange property, providing solutions to Fine's problem.

A note on a result of Bateman and Chowla

CODECA', Paolo;NAIR, Mohan K.
2000

Abstract

In 1961, answering a problem proposed by N.J.Fine, Besicovitch constructed an example of a non trivial real continous function f on [0,1] which is odd with respect to the point 1/2 and with the property that the sum of the values of the function in the points a/n, with a=1,...,n is equal to 0 for every natural n. Bateman and Chowla in 1963 pointed out that more explicit function (trigonometric cosine series with coefficients depending on Liouville's function and Moebius function) share the same property.In this paper we show that a class of functions arising as formal limits of a certain finite minimizing problem also have this strange property, providing solutions to Fine's problem.
2000
Codeca', Paolo; Nair, Mohan K.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11392/521098
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