Prime Numbers

8 Chapter 1 PRIMES!
1.1.4 Asymptotic relations and order nomenclature
At this juncture, in anticipation of many more asymptotic density results
and computational complexity estimates, we establish asymptotic relation
nomenclature for the rest of the book. When we intend
f(N) ∼ g(N)
to be read “f is asymptotic to g as N goes to infinity,” we mean that a certain
limit exists and has value unity:
When we say
lim f(N)/g(N) =1.
N→∞
f(N) =O(g(N)),
to be read “f is big-O of g,” we mean that f is bounded in this sense: There
exists a positive number C such that for all N, or for all N in a specified set,
|f(N)| ≤C|g(N)|.
The “little-o” notation can be used when one function seriously dominates
another; i.e., we say
f(N) =o(g(N))
to mean that
lim f(N)/g(N) =0.
N→∞
Some examples of the notation are in order. Since π(x), the number of
primes not exceeding x, is clearly less than x for any positive x, we can say
π(x) =O(x).
On the other hand, it is not so clear, and in fact takes some work to prove
(see Exercises 1.11 and 1.13 for two approaches), that
π(x) =o(x). (1.1)
Equation (1.1) can be interpreted as the assertion that at very high levels the
primes are sparsely distributed, and get more sparsely distributed the higher
one goes. If A is a subset of the natural numbers and A(x) denotes the number
of members of A that do not exceed x, theniflimx→∞ A(x)/x = d, we call d
the asymptotic density of the set A. Thus equation (1.1) asserts that the set
of primes has asymptotic density 0. Note that not all subsets of the natural
numbers possess an asymptotic density; that is, the limit in the definition may
not exist. As just one example, take the set of numbers with an even number
of decimal digits.
Throughout the book, when we speak of computational complexity of
algorithms we shall stay almost exclusively with “O” notation, even though