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

n= r=0 d r (n)q r avecd r (n){0,...,q-1}

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(n)= r=0 fd r (n).

Soit p(x)=α 0 x β 0 ++α d x β d avec α 0 ,α 1 ,...,α d ,, α 0 >0, β 0 >>β d 1 et au moins un β i . Un tel p est appelé pseudo-polynôme.

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

nN fp(n)=μ f Nlog q (p(N))+NF f,β 0 log q (p(N))+𝒪N 1-ε ,

μ f est la moyenne des valeurs de f et F f,β 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

n= r=0 d r (n)q r withd 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 fd r (n).

Let p(x)=α 0 x β 0 ++α d x β d with α 0 ,α 1 ,...,α d ,, α 0 >0, β 0 >>β d 1 and at least one β i . 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

nN fp(n)=μ f Nlog q (p(N))+NF f,β 0 log q (p(N))+𝒪N 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.

DOI : 10.5802/jtnb.791
Classification : 11N37 11A63
Mots clés : q additive function, pseudo-polynomial
Madritsch, Manfred G. 1

1 Department for Analysis and Computational Number Theory Graz University of Technology 8010 Graz, Austria
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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, Tome 24 (2012) no. 1, pp. 153-171. doi : 10.5802/jtnb.791. http://archive.numdam.org/articles/10.5802/jtnb.791/

[1] H. Delange, Sur la fonction sommatoire de la fonction“somme des chiffres”. Enseignement Math. (2) 21 (1975), no. 1, 31–47. | MR | Zbl

[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. | MR | Zbl

[3] A. O. Gelʼfond, Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arith. 13 (1967/1968), 259–265. | MR | Zbl

[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. | MR | Zbl

[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. | MR

[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. | MR | Zbl

[7] H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. | MR

[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. | MR | Zbl

[9] E. Krätzel, Lattice points. Mathematics and its Applications (East European Series), vol. 33, Kluwer Academic Publishers Group, Dordrecht, 1988. | MR | Zbl

[10] C. Mauduit and J. Rivat, Propriétés q-multiplicatives de la suite n c , c>1. Acta Arith. 118 (2005), no. 2, 187–203. | MR

[11] C. Mauduit and J. Rivat, La somme des chiffres des carrés. Acta Math. 203 (2009), no. 1, 107–148. | MR

[12] Y. Nakai and I. Shiokawa, A class of normal numbers. Japan. J. Math. (N.S.) 16 (1990), no. 1, 17–29. | MR | Zbl

[13] M. Peter, The summatory function of the sum-of-digits function on polynomial sequences. Acta Arith. 104 (2002), no. 1, 85–96. | MR

[14] I. Shiokawa, On the sum of digits of prime numbers. Proc. Japan Acad. 50 (1974), 551–554. | MR | Zbl

[15] J. M. Thuswaldner, The sum of digits function in number fields. Bull. London Math. Soc. 30 (1998), no. 1, 37–45. | MR | Zbl

[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. | MR | Zbl

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