The differentiability of Riemann’s functions
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- by A. Smith PDF
- Proc. Amer. Math. Soc. 34 (1972), 463-468 Request permission
Correction: Proc. Amer. Math. Soc. 89 (1983), 567-568.
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
- G. H. Hardy, Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc. 17 (1916), no. 3, 301–325. MR 1501044, DOI 10.1090/S0002-9947-1916-1501044-1
- Joseph Gerver, The differentiability of the Riemann function at certain rational multiples of $\pi$, Amer. J. Math. 92 (1970), 33–55. MR 265525, DOI 10.2307/2373496
- Richard Bellman, A brief introduction to theta functions, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1961. MR 0125252, DOI 10.1017/s0025557200044491
- Edmund Landau, Elementary number theory, Chelsea Publishing Co., New York, N.Y., 1958. Translated by J. E. Goodman. MR 0092794
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 34 (1972), 463-468
- MSC: Primary 26A27
- DOI: https://doi.org/10.1090/S0002-9939-1972-0308337-4
- MathSciNet review: 0308337