Geodesic
Some Normal Numbers Generated by Arithmetic Functions
Canadian mathematical bulletin, Tome 58 (2015) no. 1, pp. 160-173

Voir la notice de l'article provenant de la source Cambridge University Press

Let $g\,\ge \,2$ . A real number is said to be $g$ -normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\varphi$ denote Euler’s totient function, let $\sigma$ be the sum-of-divisors function, and let $\lambda$ be Carmichael’s lambda-function. We show that if $f$ is any function formed by composing $\varphi$ , $\sigma$ , or $\lambda$ , then the number $$0.f\left( 1 \right)f\left( 2 \right)f\left( 3 \right)\,.\,.\,.$$ obtained by concatenating the base $g$ digits of successive $f$ -values is $g$ -normal. We also prove the same result if the inputs 1,2,3....are replaced with the primes 2, 3, 5.... The proof is an adaptation of a method introduced by Copeland and Erdõs in 1946 to prove the 10-normality of 0:235711131719...
DOI : 10.4153/CMB-2014-047-2
Mots-clés : 11K16, 11A63, 11N25, 11N37, normal number, Euler function, sum-of-divisors function, Carmichael lambda-function, Champernowne’s number 
Pollack, Paul; Vandehey, Joseph. Some Normal Numbers Generated by Arithmetic Functions. Canadian mathematical bulletin, Tome 58 (2015) no. 1, pp. 160-173. doi: 10.4153/CMB-2014-047-2
@article{10_4153_CMB_2014_047_2,
     author = {Pollack, Paul and Vandehey, Joseph},
     title = {Some {Normal} {Numbers} {Generated} by {Arithmetic} {Functions}},
     journal = {Canadian mathematical bulletin},
     pages = {160--173},
     year = {2015},
     volume = {58},
     number = {1},
     doi = {10.4153/CMB-2014-047-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2014-047-2/}
}
TY  - JOUR
AU  - Pollack, Paul
AU  - Vandehey, Joseph
TI  - Some Normal Numbers Generated by Arithmetic Functions
JO  - Canadian mathematical bulletin
PY  - 2015
SP  - 160
EP  - 173
VL  - 58
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2014-047-2/
DO  - 10.4153/CMB-2014-047-2
ID  - 10_4153_CMB_2014_047_2
ER  - 
%0 Journal Article
%A Pollack, Paul
%A Vandehey, Joseph
%T Some Normal Numbers Generated by Arithmetic Functions
%J Canadian mathematical bulletin
%D 2015
%P 160-173
%V 58
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2014-047-2/
%R 10.4153/CMB-2014-047-2
%F 10_4153_CMB_2014_047_2

[1] [1] Bassily, N. L., Kátai, I., and Wijsmuller, M., On the prime power divisors of the iterates of the Euler-' function. Publ. Math. Debrecen 55 (1999), no. 1–2. 17–32. Google Scholar

[2] [2] Besicovitch, A. S., The asymptotic distribution of the numerals in the decimal representation of the squares of the natural numbers. Math. Z. 39 (1935), no. 1, 146–156. Google Scholar | DOI

[3] [3] Borel, E., Les probabilit´es d´enombrables et leurs applications arithm´etiques. Supplemento di rend. circ. Mat. Palermo 27 (1909), 247–271. Google Scholar

[4] [4] Champernowne, D. G., The construction of decimals normal in the scale of ten. J. London Math. Soc. 3 (1933), 254–260. Google Scholar | DOI

[5] [5] Copeland, A. H. and P. Erdʺos, Note on normal numbers. Bull. Amer. Math. Soc. 52 (1946), 857–860. Google Scholar | DOI

[6] [6] Davenport, H. andErdʺos, P., Note on normal decimals. Canad. J. Math. 4 (1952), 58–63. Google Scholar | DOI

[7] [7] De Koninck, J.-M. and Kátai, I., On a problem on normal numbers raised by Igor Shparlinski. Bull. Aust. Math. Soc. 84 (2011), no. 2, 337–349. Google Scholar | DOI

[8] [8] De Koninck, J.-M. and Kátai, I., Using large prime divisors to construct normal numbers. Ann. Univ. Sci. Budapest. Sect. Comput. 39 (2013), 45–62. Google Scholar

[9] [9] Erdʺos, P., Granville, A., Pomerance, C., and Spiro, C., On the normal behavior of the iterates of some arithmetic functions. In: Analytic number theory (Allerton Park, IL, 1989), Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990, pp. 165–204.. Google Scholar

[10] [10] Friedlander, J. B., Pomerance, C., and Shparlinski, I. E., Period of the power generator and small values of Carmichael's function. Math. Comp.70 (2001), no. 236, 1591–1605.(electronic); corrigendum 71 (2002), 1803–1806. Google Scholar | DOI

[11] [11] Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers. Sixth ed., Oxford University Press, Oxford, 2008. Google Scholar

[12] [12] Kurlberg, P. and Rudnick, Z., On quantum ergodicity for linear maps of the torus. Comm. Math. Phys. 222 (2001), no. 1, 201–227. Google Scholar | DOI

[13] [13] Luca, F. and Pollack, P., An arithmetic function arising from Carmichael's conjecture. J. Th´eor. Nombres Bordeaux 23 (2011), no. 3, 697–714. Google Scholar | DOI

[14] [14] Luca, F. and Pomerance, C., Irreducible radical extensions and Euler-function chains. In: Combinatorial number theory, de Gruyter, Berlin, 2007, pp. 351–361.. Google Scholar

[15] [15] Madritsch, M. G., Thuswaldner, J. M., and Tichy, R. F., Normality of numbers generated by the values of entire functions. J. Number Theory 128 (2008), no. 5, 1127–1145. Google Scholar | DOI

[16] [16] Nakai, Y. and Shiokawa, I., Normality of numbers generated by the values of polynomials at primes. Acta Arith. 81 (1997), no. 4, 345–356. Google Scholar

[17] [17] Pollack, P. and Thompson, L., Practical pretenders. Publ.Math. Debrecen 82 (2013), no. 3–4. 651–667. Google Scholar | DOI

[18] [18] Vandehey, J., The normality of digits in almost constant additive functions. Monatsh. Math. 171 (2013), no. 3–4. 481–497. Google Scholar | DOI

Cité par Sources :