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Transactions of the American Mathematical Society
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
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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.


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Additional Information

Michael Drmota
Affiliation: Department of Geometry, Technical University of Vienna, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
Email: michael.drmota@tuwien.ac.at

Mariusz Skalba
Affiliation: Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland
Email: skalba@mimuw.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02277-1
PII: S 0002-9947(99)02277-1
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