Exponential sums related to binomial coefficient parity
HTML articles powered by AMS MathViewer
- by Alan H. Stein PDF
- Proc. Amer. Math. Soc. 80 (1980), 526-530 Request permission
Abstract:
Let $\alpha (n)$ be the number of 1’s in the binary expansion of n, $z > 0$ and ${\phi _z}(x) = {\Sigma _{n < x}}{z^{\alpha (n)}}$. Let ${\theta _z} = (\log (1 + z))/\log 2,a(z) = \lim \inf {x^{ - {\theta _z}}}{\phi _z}(x),b(z) = \lim \sup {x^{ - {\theta _z}}}{\phi _z}(x)$ . Then $0 < a(z) \leqslant 1 \leqslant b(z) < 2$. Furthermore, if $z \ne 1$, then $a(z) < b(z)$.References
- Richard Bellman and Harold N. Shapiro, On a problem in additive number theory, Ann. of Math. (2) 49 (1948), 333–340. MR 23864, DOI 10.2307/1969281
- N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54 (1947), 589–592. MR 23257, DOI 10.2307/2304500 J. W. L. Glaisher, On the residue of a binominal-theorem coefficient with respect to a prime modulus, Quart. J. Pure Appl. Math. 30 (1899), 150-156.
- Heiko Harborth, Number of odd binomial coefficients, Proc. Amer. Math. Soc. 62 (1976), no. 1, 19–22 (1977). MR 429714, DOI 10.1090/S0002-9939-1977-0429714-1
- M. D. McIlroy, The number of $1$’s in binary integers: bounds and extremal properties, SIAM J. Comput. 3 (1974), 255–261. MR 436687, DOI 10.1137/0203020
- Alan H. Stein and Ivan E. Stux, A mean value theorem for binary digits, Pacific J. Math. 75 (1978), no. 2, 565–577. MR 567133
- Kenneth B. Stolarsky, Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math. 32 (1977), no. 4, 717–730. MR 439735, DOI 10.1137/0132060
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 80 (1980), 526-530
- MSC: Primary 10A21; Secondary 05A10
- DOI: https://doi.org/10.1090/S0002-9939-1980-0581019-4
- MathSciNet review: 581019