Vol 211, No 3 (2020)

The sums $S_q(N)$ are defined by the equality $S_q(N)=s_q(1)+…+s_q(N-1)$ for all positive integers $N$ and $q\ge2$, 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.

Suppose that a curvilinear three-web is given by the equation $F(x,y,z)=0$. A specific structure of the derivatives of the function $F$ is established that characterizes regular three-webs. This makes it possible to list all regular three-webs formed by the Cartesian net and a family of circles, and also by the Cartesian net and a family of second-order curves. Bibliography: 4 titles.

We prove that a finite group acting by birational automorphisms of a nontrivial Severi-Brauer surface over a field of characteristic zero contains a normal abelian subgroup of index at most $3$. Also, we find an explicit bound for the orders of such finite groups in the case when the base field contains all roots of $1$. Bibliography: 25 titles.