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.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
