Abstract:
The sums Sq(N) are defined by the equality Sq(N)=sq(1)+⋯+sq(N−1) for all positive integers N and q⩾2, where sq(n) is the sum of digits of the integer n written in the system with base q. In 1975 Delange generalised Trollope's formula and proved that Sq(N)/N−(q−1)/2⋅logqN=−1/2⋅fq(q{logqN}−1), where fq(x)=(q−1)logqx+Dq(x)/x and Dq is the continuous nowhere differentiable Delange function. We find global extrema of fq and, using this, obtain a precise bound for the difference Sq(N)/N−(q−1)/2⋅logqN. In the case q=2 this becomes the bound for binary sums proved by Krüppel in 2008 and also earlier by other authors. We also evaluate the global extrema of some other continuous nowhere differentiable functions. We introduce the natural concave hull of a function and prove a criterion simplifying the evaluation of this hull. Moreover, we introduce the notion of an extreme subargument of a function on a convex set. We show that all points of global maximum of the difference f−g, where the function g is strictly concave and some additional conditions hold, are extreme subarguments for f. A similar result is obtained for functions of the form v+f/w. We evaluate the global extrema and find extreme subarguments of the Delange function on the interval [0,1]. The results in the paper are illustrated by graphs and tables.
Bibliography: 16 titles.
Keywords:
Trollope-Delange formula for digital sums, continuous nowhere differentiable Delange function, global extrema of a non-differentiable function, extreme subarguments (subabscissas) of a function, natural concave hull of a function.
Citation:
O. E. Galkin, S. Yu. Galkina, “Global extrema of the Delange function, bounds for digital sums and concave functions”, Sb. Math., 211:3 (2020), 336–372
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\by O.~E.~Galkin, S.~Yu.~Galkina
\paper Global extrema of the Delange function, bounds for digital sums and concave functions
\jour Sb. Math.
\yr 2020
\vol 211
\issue 3
\pages 336--372
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Linking options:
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This publication is cited in the following 4 articles:
Anna Sappington, Vaibhav Mohanty, “Probabilistic genotype-phenotype maps reveal mutational robustness of RNA folding, spin glasses, and quantum circuits”, Phys. Rev. Research, 7:1 (2025)
O. E. Galkin, S. Yu. Galkina, A. A. Tronov, “O globalnykh ekstremumakh stepennykh funktsii Takagi”, Zhurnal SVMO, 25:2 (2023), 22–36
O. E. Galkin, S. Yu. Galkina, O. A. Mulyar, “O logarifmicheskoi gelderovosti i lokalnykh ekstremumakh stepennykh funktsii Takagi”, Zhurnal SVMO, 25:4 (2023), 223–241
V. Mohanty, S. F. Greenbury, T. Sarkany, Sh. Narayanan, K. Dingle, S. E. Ahnert, A. A. Louis, “Maximum mutational robustness in genotype–phenotype maps follows a self-similar blancmange-like curve”, J. R. Soc. Interface, 20:204 (2023), 20230169