  All the known approximations of the number of primes pi(n) not exceeding any
given integer n are derived from real-valued functions that are asymptotic to
pi(x), such as x/log x, Li(x) and Riemann's function R(x). The degree of
approximation for finite values of n is determined only heuristically, by
conjecturing upon an error term in the asymptotic relation that can be seen to
yield a closer approximation than others to the actual values of pi(n) within a
finite range of values of n. By considering the density of each of the set of
(i) all integers n, (ii) Dirichlect integers n = a+md, and (iii) Twin integers
(n, n+2), which are not divisible by any of the first k primes, we show
that---based on their respective densities---the expected number of such
integers in the initial interval (1, n) of length n non-heuristically
approximates the number of (a) primes, (b) Dirichlect primes, and (c) Twin
primes, respectively, which are less than or equal to n. We further show that,
in each case, the estimate tends to infinity.
