Differentiability of Riemann's function

Abstract

1. Introduction.In this paper we discuss on the differenti- ability of the function f(x)= , sin nx/n .Riemann proposed the problem that the function is nowhere differenti- able, [2] and [8].About the problem, J. P. Kahane [3] has investigated lacunary series.It was solved by J. Gerver [4] [5].First G. H. Hardy [6] proved that the function is not differentiable at the point $ where $ is irrational or is a rational of the form (2A/l)/2B or 2A/(4B/ 1).Later Gerver proved that f(x) is differentiable at all points (2A/ 1)=/(2B + 1) with derivative 1/2, and not differentiable at the points 2A/(2B + 1).

Locations

• Proceedings of the Japan Academy Series A Mathematical Sciences - View - PDF