Exponential sums related to binomial coefficient parity

Stein, Alan H.

doi:10.1090/S0002-9939-1980-0581019-4

Exponential sums related to binomial coefficient parity
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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 $a(z) < b(z)$.

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