The summatory function of $q$-additive functions on pseudo-polynomial sequences

Madritsch, Manfred G.

Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 1, pp. 153-171.

Le présent article étudie la fonction sommatoire de fonctions définies sur les chiffres en base $q$. En particulier, si $n$ est un entier positif, nous notons

$$\begin{array}{c}n=\sum _{r=0}^{\ell }{d}_r\left(n\right)q^r\text{avec}{d}_r\left(n\right)\in \{0,...,q-1\}\end{array}$$

son développement en base $q$. Nous disons qu’une fonction $f$ est strictement $q$-additive si, pour une valeur donnée, elle agit uniquement sur les chiffres de sa représentation, i.e.,

$$f\left(n\right)=\sum _{r=0}^{\ell }f\left(d_r\left(n\right)\right).$$

Soit $p\left(x\right)={\alpha }_0{x}^{{\beta }_0}+\cdots +{\alpha }_d{x}^{{\beta }_d}$ avec ${\alpha }_0,{\alpha }_1,...,{\alpha }_d,\in ℝ$, ${\alpha }_0>0$, ${\beta }_0>\cdots >{\beta }_d\ge 1$ et au moins un ${\beta }_i\notin \mathbb{Z}$. Un tel $p$ est appelé pseudo-polynôme.

Le but est de prouver que pour $f$ une fonction $q$-additive, il existe un $\epsilon >0$ tel que

$$\begin{array}{c}\sum _{n\le N}f\left(⌊p\left(n\right)⌋\right)={\mu }_{f}N{log}_q\left(p\left(N\right)\right)\hfill \\ \hfill +N{F}_{f,{\beta }_0}\left({log}_q\left(p\left(N\right)\right)\right)+𝒪\left(N^{1-\epsilon }\right),\end{array}$$

où ${\mu }_f$ est la moyenne des valeurs de $f$ et $F_{f,{\beta }_0}$ est une fonction $1$-périodique dérivable nulle part.

Ce résulat est motivé par des résultats de Nakai et Shiokawa et de Peter.

The present paper deals with the summatory function of functions acting on the digits of an $q$-ary expansion. In particular let $n$ be a positive integer, then we call

$$\begin{array}{c}n=\sum _{r=0}^{\ell }{d}_r\left(n\right)q^r\text{with}{d}_r\left(n\right)\in \{0,...,q-1\}\end{array}$$

its $q$-ary expansion. We call a function $f$ strictly $q$-additive, if for a given value, it acts only on the digits of its representation, i.e.,

$$f\left(n\right)=\sum _{r=0}^{\ell }f\left(d_r\left(n\right)\right).$$

Let $p\left(x\right)={\alpha }_0{x}^{{\beta }_0}+\cdots +{\alpha }_d{x}^{{\beta }_d}$ with ${\alpha }_0,{\alpha }_1,...,{\alpha }_d,\in ℝ$, ${\alpha }_0>0$, ${\beta }_0>\cdots >{\beta }_d\ge 1$ and at least one ${\beta }_i\notin \mathbb{Z}$. Then we call $p$ a pseudo-polynomial.

The goal is to prove that for a $q$-additive function $f$ there exists an $\epsilon >0$ such that

$$\begin{array}{c}\sum _{n\le N}f\left(⌊p\left(n\right)⌋\right)={\mu }_{f}N{log}_q\left(p\left(N\right)\right)\hfill \\ \hfill +N{F}_{f,{\beta }_0}\left({log}_q\left(p\left(N\right)\right)\right)+𝒪\left(N^{1-\epsilon }\right),\end{array}$$

where ${\mu }_f$ is the mean of the values of $f$ and $F_{f,{\beta }_0}$ is a $1$-periodic nowhere differentiable function.

This result is motivated by results of Nakai and Shiokawa and Peter.

Classification : 11N37 11A63
Mots clés : q additive function, pseudo-polynomial

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