The differentiability of Riemann's functions

A. Smith
1972 Proceedings of the American Mathematical Society  
The function g{x)= ]>£Li (sin np^x/irp-), 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.
doi:10.1090/s0002-9939-1972-0308337-4 fatcat:rt5azkdkyndh5j2rvltmxb527q