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

@article{Madritsch2012,
abstract = {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\{gather*\} n=\sum \_\{r=0\}^\ell d\_r(n)q^r\quad \text\{with\}\quad d\_r(n)\in \lbrace 0,\ldots ,q-1\rbrace \end\{gather*\}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.,\[ f(n)=\sum \_\{r=0\}^\ell f\left(d\_r(n)\right). \]Let $p(x)=\alpha _0x^\{\beta _0\}+\cdots +\alpha _dx^\{\beta _d\}$ with $\alpha _0,\alpha _1,\ldots ,\alpha _d,\in \mathbb\{R\}$, $\alpha _0>0$, $\beta _0>\cdots >\beta _d\ge 1$ 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 $\varepsilon >0$ such that\begin\{multline*\} \sum \_\{n\le N\}f\left(\left\lfloor p(n)\right\rfloor \right) =\mu \_fN\log \_q(p(N))\\ +NF\_\{f,\beta \_0\}\left(\log \_q(p(N))\right) +\mathcal\{O\}\left(N^\{1-\varepsilon \}\right), \end\{multline*\}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.},
affiliation = {Department for Analysis and Computational Number Theory Graz University of Technology 8010 Graz, Austria},
author = {Madritsch, Manfred G.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {q additive function; pseudo-polynomial; q-additive function},
language = {eng},
month = {3},
number = {1},
pages = {153-171},
publisher = {Société Arithmétique de Bordeaux},
title = {The summatory function of $q$-additive functions on pseudo-polynomial sequences},
url = {http://eudml.org/doc/251059},
volume = {24},
year = {2012},
}

TY - JOUR
AU - Madritsch, Manfred G.
TI - The summatory function of $q$-additive functions on pseudo-polynomial sequences
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/3//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 1
SP - 153
EP - 171
AB - 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{gather*} n=\sum _{r=0}^\ell d_r(n)q^r\quad \text{with}\quad d_r(n)\in \lbrace 0,\ldots ,q-1\rbrace \end{gather*}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.,\[ f(n)=\sum _{r=0}^\ell f\left(d_r(n)\right). \]Let $p(x)=\alpha _0x^{\beta _0}+\cdots +\alpha _dx^{\beta _d}$ with $\alpha _0,\alpha _1,\ldots ,\alpha _d,\in \mathbb{R}$, $\alpha _0>0$, $\beta _0>\cdots >\beta _d\ge 1$ 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 $\varepsilon >0$ such that\begin{multline*} \sum _{n\le N}f\left(\left\lfloor p(n)\right\rfloor \right) =\mu _fN\log _q(p(N))\\ +NF_{f,\beta _0}\left(\log _q(p(N))\right) +\mathcal{O}\left(N^{1-\varepsilon }\right), \end{multline*}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.
LA - eng
KW - q additive function; pseudo-polynomial; q-additive function
UR - http://eudml.org/doc/251059
ER -