Portal de Periódicos da CAPES

The Differentiability of Riemann's Function

1972; American Mathematical Society; Volume: 34; Issue: 2 Linguagem: Inglês

10.2307/2038391

1088-6826

Arthur L. Smith,

Mathematical and Theoretical Analysis

The functiong(x)= :P%-= (sin 7rp2X/7rp2), thought by Riemann to be nowhere differentiable, is shown to be differentiable only at rational points expressible as the ratio of odd integers. The proof depends on properties of Gaussian sums, and these properties enable us to give a complete discussion of the possible existence of left and right derivatives at any point. Introduction. Riemann asserted that the continuous function g(x) = E sin 7Tp x p=l '7P2 is nowhere differentiable, and Hardy [1] proved that g has no finite derivative at irrational points nor at rational points of the form 2p/(4q+ 1) or (2p+1)/(4q+2). Recently interest in this problem has be'n revived because Gerver [2] proved that g has a derivative 2 at points of the form (2p+1)/(2q+1). He also proved that g has no finite derivative at points of the form (2p + 1)/2' with n ?1. We now extend the results to the remaining cases, showing also the existence of finite leftand right-hand derivatives at certain rationals, and proving that these derivatives exist at all rationals if we allow the values ? xo. Since Gerver's proof is extremely long and Hardy obtains his results indirectly we give a completely elementary and fairly short proof of all the above assertions. For convenience we work with the function (1) f (x) = x + 2g(x) = x + 2 2 so that, for example, we shall verify thatf'(x) exists and is zero when x is of the form (2p + 1)/(2q + 1). The following lemma is all that is required to obtain expansions of f(x) about a rational point x, which using properties of Gaussian sums reveal the properties of the derivatives. Received by the editors May 24, 1971. AMS 1970 subject classiflcations. Primary 26A27; Secondary IOG05, 40A05, 42A20.