The differentiability of Riemann’s functions

Abstract:

The function $g(x) = \sum \nolimits _{p = 1}^\infty {(\sin \pi {p^2}x/\pi {p^2})}$, 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.