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

• [1] Department for Analysis and Computational Number Theory Graz University of Technology 8010 Graz, Austria
• Volume: 24, Issue: 1, page 153-171
• ISSN: 1246-7405

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## Abstract

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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$\begin{array}{c}n=\sum _{r=0}^{\ell }{d}_{r}\left(n\right){q}^{r}\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}{d}_{r}\left(n\right)\in \left\{0,...,q-1\right\}\end{array}$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 ${\beta }_{i}\notin ℤ$. 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.

## How to cite

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Madritsch, Manfred G.. "The summatory function of $q$-additive functions on pseudo-polynomial sequences." Journal de Théorie des Nombres de Bordeaux 24.1 (2012): 153-171. <http://eudml.org/doc/251059>.

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 $\alpha _0,\alpha _1,\ldots ,\alpha _d,\in \mathbb\{R\}$, $\alpha _0&gt;0$, $\beta _0&gt;\cdots &gt;\beta _d\ge 1$ 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 $\varepsilon &gt;0$ 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},
journal = {Journal de Théorie des Nombres de Bordeaux},
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
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 $\alpha _0,\alpha _1,\ldots ,\alpha _d,\in \mathbb{R}$, $\alpha _0&gt;0$, $\beta _0&gt;\cdots &gt;\beta _d\ge 1$ 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 $\varepsilon &gt;0$ 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
UR - http://eudml.org/doc/251059
ER -

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