Global extrema of the Delange function, bounds for digital sums and concave functions

O. E. Galkin ; S. Yu. Galkina

Sbornik. Mathematics, Tome 211 (2020) no. 3, pp. 336-372

Résumé

The sums $S_q(N)$ are defined by the equality $S_q(N)=s_q(1)+\dots+s_q(N-1)$ for all positive integers $N$ and $q\geqslant2$, where $s_q(n)$ 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 $S_q(N)/N-({q-1})/2\cdot\log_qN=-1/2\cdot f_q(q^{\{\log_q N\}-1})$, where $f_q(x)=(q-1)\log_q x+D_q(x)/x$ and $D_q$ is the continuous nowhere differentiable Delange function. We find global extrema of $f_q$ and, using this, obtain a precise bound for the difference $S_q(N)/N-(q-1)/2\cdot\log_qN$. 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.

Détail

Keywords: Trollope-Delange formula for digital sums, continuous nowhere differentiable Delange function, global extrema of a non-differentiable function, natural concave hull of a function.
Mots-clés : extreme subarguments (subabscissas) of a function