Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Rarified sums of the Thue-Morse sequence

Authors: Michael Drmota and Mariusz Skalba
Journal: Trans. Amer. Math. Soc. 352 (2000), 609-642
MSC (1991): Primary 11B85; Secondary 11A63
Published electronically: August 10, 1999
MathSciNet review: 1491859
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $q$ be an odd number and $S_{q,0}(n)$ the difference between the number of $k<n$, $k\equiv 0\bmod\,q$, with an even binary digit sum and the corresponding number of $k<n$, $k\equiv 0\bmod\,q$, with an odd binary digit sum. A remarkable theorem of Newman says that $S_{3,0}(n)>0$ for all $n$. In this paper it is proved that the same assertion holds if $q$ is divisible by 3 or $q=4^N+1$. On the other hand, it is shown that the number of primes $q\le x$ with this property is $o(x/\log x)$. Finally, analoga for ``higher parities'' are provided.

References [Enhancements On Off] (What's this?)

  • 1. J. Coquet, A summation formula related to the binary digits, Invent. Math. 73 (1983), 107-115. MR 85c:11012
  • 2. M. Drmota and M. Ska{\l}ba, Sign-changes of the Thue-Morse sequence and Dirichlet L-series, manuscripta math. 86 (1995), 519-541. MR 96b:11027
  • 3. J. M. Dumont, Discrépance des progressions arithméthiques dans la suite de Morse, C. R. Acad. Sci. Paris, Série I 297 (1983), 145-148. MR 85f:11058
  • 4. P. Erd\H{o}s, Bemerkungen zu einer Aufgabe von Elementen, Archiv Math. 27 (1976), 159-163. MR 53:7969
  • 5. S. Goldstein, K. A. Kelly, and E. R. Speer, The fractal structure of rarefied sums of the Thue-Morse sequence, J. Number Th. 42 (1992), 1-19. MR 93m:11020
  • 6. P. J. Grabner, A note on the parity of the sum-of-digits function, Actes $30^{\rm e}$ Séminaire Lotharingien de Combinatoire, 1993, 35-42. MR 95k:11125
  • 7. P. J. Grabner, T. Herendi, and R. F. Tichy, Fractal digital sums and codes, Appl. Algebra Engin. Comm. Comput. 1 (1997), 33-39. CMP 97:17
  • 8. H. Leinfellner, Thesis, TU Wien, 1998.
  • 9. M. Morse, Reccurent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc. 22 (1921), 84-100.
  • 10. D. J. Newman, On the number of binary digits in a multiple of three, Proc. Am. Math. Soc. 21 (1969), 719-721. MR 39:5466
  • 11. K. Prachar, `Primzahlverteilung,' Springer, Berlin, 1957. MR 19:393b
  • 12. I. M. Vinogradov, `Elemente der Zahlentheorie,' Oldenbourg, München, 1956.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11B85, 11A63

Retrieve articles in all journals with MSC (1991): 11B85, 11A63

Additional Information

Michael Drmota
Affiliation: Department of Geometry, Technical University of Vienna, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria

Mariusz Skalba
Affiliation: Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland

Received by editor(s): July 6, 1995
Received by editor(s) in revised form: December 2, 1997
Published electronically: August 10, 1999
Additional Notes: This work was supported by the Austrian Science Foundation, grant Nr. M 00233–MAT
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society