The joint distribution of the digits of certain integer s-tuples

Abstract

Let B(n) denote the number of digits 1 in the representation of a natural number n in the binary scale. For example, since 5 in the binary scale is 101, we have B(5) = 2. It is well known that for most integers n, the number B(n) is about half the total number of digits in the binary representation of n, so that B(n) is roughly equal to $\frac{1} {2}v$, where

$$v = v(n) = _2 \log n,$$

(1.1)

and where ₂log is the logarithm to the base 2. In fact it follows from the Central Limit Theorem of probability theory that the numbers n with

$$\frac{{B(n) - \frac{1} {2}v}} {{\sqrt v }} \leqq \xi$$

(1.2)

have density

$$\phi (\xi ) = \sqrt {\frac{2} {\pi }\int\limits_{ - \infty }^\xi {e^{ - 2t^2 } dt:} }$$

(1.3)

Here and always we say that a set S of natural numbers has density ψ if the number N(S, x) of numbers n ∈ S, n ≦ x satisfies the asymptotic relation N(S, x) ∼ ψx as x → ∞.