Rarified sums of the Thue-Morse sequence

Drmota, Michael; Skałba, Mariusz

doi:10.1090/S0002-9947-99-02277-1

Rarified sums of the Thue-Morse sequence
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by Michael Drmota and Mariusz Skałba

Trans. Amer. Math. Soc. 352 (2000), 609-642

DOI: https://doi.org/10.1090/S0002-9947-99-02277-1

Published electronically: August 10, 1999

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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.

References

J. Coquet, A summation formula related to the binary digits, Invent. Math. 73 (1983), no. 1, 107–115. MR 707350, DOI 10.1007/BF01393827
Michael Drmota and Mariusz Skałba, Sign-changes of the Thue-Morse fractal function and Dirichlet $L$-series, Manuscripta Math. 86 (1995), no. 4, 519–541. MR 1324686, DOI 10.1007/BF02568009
Jean-Marie Dumont, Discrépance des progressions arithmétiques dans la suite de Morse, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 3, 145–148 (French, with English summary). MR 725391
P. Erdős, Bemerkungen zu einer Aufgabe (Elem. Math. 26 (1971), 43) by G. Jaeschke, Arch. Math. (Basel) 27 (1976), no. 2, 159–163. MR 404166, DOI 10.1007/BF01224655
Sheldon Goldstein, Kevin A. Kelly, and Eugene R. Speer, The fractal structure of rarefied sums of the Thue-Morse sequence, J. Number Theory 42 (1992), no. 1, 1–19. MR 1176416, DOI 10.1016/0022-314X(92)90104-W
Peter J. Grabner, A note on the parity of the sum-of-digits function, Séminaire Lotharingien de Combinatoire (Gerolfingen, 1993) Prépubl. Inst. Rech. Math. Av., vol. 1993/34, Univ. Louis Pasteur, Strasbourg, 1993, pp. 35–42. MR 1312627
P. J. Grabner, T. Herendi, and R. F. Tichy, Fractal digital sums and codes, Appl. Algebra Engin. Comm. Comput. 1 (1997), 33–39.
H. Leinfellner, Thesis, TU Wien, 1998.
M. Morse, Reccurent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc. 22 (1921), 84–100.
Donald J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969), 719–721. MR 244149, DOI 10.1090/S0002-9939-1969-0244149-8
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
I. M. Vinogradov, ‘Elemente der Zahlentheorie,’ Oldenbourg, München, 1956.