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Proceedings of the American Mathematical Society
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
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0308337-4
PII: S 0002-9939(1972)0308337-4
Keywords: Riemann function, nondifferentiability, Poisson summation formula, Gaussian sums
Article copyright: © Copyright 1972 American Mathematical Society