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

Manfred G. Madritsch[1]

  • [1] Department for Analysis and Computational Number Theory Graz University of Technology 8010 Graz, Austria

Journal de Théorie des Nombres de Bordeaux (2012)

  • Volume: 24, Issue: 1, page 153-171
  • ISSN: 1246-7405

Abstract

top
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 n = ∑ r = 0 ℓ d r ( n ) q r with d r ( n ) ∈ { 0 , ... , q - 1 } 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 ) = ∑ r = 0 ℓ f d r ( n ) . Let p ( x ) = α 0 x β 0 + ⋯ + α d x β d with α 0 , α 1 , ... , α d , ∈ ℝ , α 0 > 0 , β 0 > ⋯ > β d ≥ 1 and at least one β i ∉ ℤ . Then we call p a pseudo-polynomial.The goal is to prove that for a q -additive function f there exists an ε > 0 such that ∑ n ≤ N f p ( n ) = μ f N log q ( p ( N ) ) + N F f , β 0 log q ( p ( N ) ) + 𝒪 N 1 - ε , where μ f is the mean of the values of f and F f , β 0 is a 1 -periodic nowhere differentiable function.This result is motivated by results of Nakai and Shiokawa and Peter.

How to cite

top

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>.

@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 $\alpha _0,\alpha _1,\ldots ,\alpha _d,\in \mathbb\{R\}$, $\alpha _0&gt;0$, $\beta _0&gt;\cdots &gt;\beta _d\ge 1$ 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},
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 $\alpha _0,\alpha _1,\ldots ,\alpha _d,\in \mathbb{R}$, $\alpha _0&gt;0$, $\beta _0&gt;\cdots &gt;\beta _d\ge 1$ 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
KW - q additive function; pseudo-polynomial; q-additive function
UR - http://eudml.org/doc/251059
ER -

References

top
  1. H. Delange, Sur la fonction sommatoire de la fonction“somme des chiffres”. Enseignement Math. (2) 21 (1975), no. 1, 31–47. Zbl0306.10005MR379414
  2. P. Flajolet, P. Grabner, P. Kirschenhofer, H. Prodinger, and R. F. Tichy, Mellin transforms and asymptotics: digital sums. Theoret. Comput. Sci. 123 (1994), no. 2, 291–314. Zbl0788.44004MR1256203
  3. A. O. Gelʼfond, Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arith. 13 (1967/1968), 259–265. Zbl0155.09003MR220693
  4. B. Gittenberger and J. M. Thuswaldner, The moments of the sum-of-digits function in number fields. Canad. Math. Bull. 42 (1999), no. 1, 68–77. Zbl1011.11009MR1695870
  5. P. J. Grabner and H.-K. Hwang, Digital sums and divide-and-conquer recurrences: Fourier expansions and absolute convergence. Constr. Approx. 21 (2005), no. 2, 149–179. Zbl1088.11063MR2107936
  6. P. J. Grabner, P. Kirschenhofer, H. Prodinger, and R. F. Tichy, On the moments of the sum-of-digits function. Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, pp. 263–271. Zbl0797.11012MR1271366
  7. H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. Zbl1059.11001MR2061214
  8. P. Kirschenhofer, On the variance of the sum of digits function. Number-theoretic analysis (Vienna, 1988–89), Lecture Notes in Math., vol. 1452, Springer, Berlin, 1990, pp. 112–116. Zbl0714.11005MR1084640
  9. E. Krätzel, Lattice points. Mathematics and its Applications (East European Series), vol. 33, Kluwer Academic Publishers Group, Dordrecht, 1988. Zbl0675.10031MR998378
  10. C. Mauduit and J. Rivat, Propriétés q -multiplicatives de la suite ⌊ n c ⌋ , c &gt; 1 . Acta Arith. 118 (2005), no. 2, 187–203. Zbl1082.11058MR2141049
  11. C. Mauduit and J. Rivat, La somme des chiffres des carrés. Acta Math. 203 (2009), no. 1, 107–148. MR2545827
  12. Y. Nakai and I. Shiokawa, A class of normal numbers. Japan. J. Math. (N.S.) 16 (1990), no. 1, 17–29. Zbl0708.11037MR1064444
  13. M. Peter, The summatory function of the sum-of-digits function on polynomial sequences. Acta Arith. 104 (2002), no. 1, 85–96. Zbl1027.11070MR1913736
  14. I. Shiokawa, On the sum of digits of prime numbers. Proc. Japan Acad. 50 (1974), 551–554. Zbl0301.10047MR369238
  15. J. M. Thuswaldner, The sum of digits function in number fields. Bull. London Math. Soc. 30 (1998), no. 1, 37–45. Zbl0921.11051MR1479034
  16. E. C. Titchmarsh, The theory of the Riemann zeta-function, second ed. The Clarendon Press Oxford University Press, New York, 1986, Edited and with a preface by D. R. Heath-Brown. Zbl0601.10026MR882550

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.