xiv Notation

We write lcm[a1,a2,...,ak] to denote the least common multiple of the

positive integers a1,a2,...,ak. Similarly, we write gcd(a1,a2,...,ak) to de-

note the greatest common divisor of the positive integers a1,a2,...,ak; when

the context is clear, we may at times simply write (a1,a2,...,ak) instead of

gcd(a1,a2,...,ak), as well as [a1,a2,...,ak] instead of lcm[a1,a2,...,ak].

Throughout this book, log x stands for the natural logarithm of x. At

times, we also write log2 x instead of log log x and more generally, for each

integer k ≥ 3, we let logk x stand for log(logk−1 x).

For each integer k ≥ 0, we denote by

Ck[a, b] the set of functions whose

kth derivative exists and is continuous on the interval [a, b]. Thus

C0[a,

b]

stands for the set of continuous functions on [a, b].

It is often convenient to use the notations introduced by Landau, namely

o(...) and O(...), to compare the order of magnitude of functions in the

neighborhood of a point or of infinity. Unless we indicate otherwise, we

shall mean the latter.

Given two functions f and g defined on [a, ∞), where a ≥ 0 and g(x) 0,

we shall write that f(x) = O(g(x)) if there exist two constants M 0 and

x0 such that |f(x)| Mg(x) for all x ≥ x0. In particular, f(x) = O(1) if

f(x) is a bounded function. Moreover, instead of writing f(x) = O(g(x)),

we shall at times write f(x) g(x).

Thus, we have, as x → ∞,

x =

O(x2),

sin x = O(1), log x =

O(x1/10),

sin x

x

= O(1),

x4

=

O(ex),

ex

sin x =

O(ex),

2x2

+

x

3

x2,

x3

x3+x2

+ 7 = O(1).

Given two functions f and g defined on [a, ∞), where a ≥ 0 and g(x) 0,

we shall write f(x) = o(g(x)) as x → ∞ if, for each ε 0, there exists a

constant x0 = x0(ε) such that |f(x)| εg(x) for all x ≥ x0.

Thus, we have

1

x

= o(1), sin x = o(x), log x = o(x),

sin 1/x

1/x

= 1 + o(1),

x4

=

o(ex), xex

sin x =

o(x2ex).