Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The differentiability of Riemann's functions


Author: A. Smith
Journal: Proc. Amer. Math. Soc. 34 (1972), 463-468
MSC: Primary 26A27
MathSciNet review: 0308337
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A27

Retrieve articles in all journals with MSC: 26A27


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0308337-4
Keywords: Riemann function, nondifferentiability, Poisson summation formula, Gaussian sums
Article copyright: © Copyright 1972 American Mathematical Society