8 1. Arithmetic Functions

independently of the parameter a, then the estimate is said to be uniform in

a. Neglect of uniformity is a recognized source of error in number theoretical

arguments. The case where C is independent of a while x0 depends on a in

some (unobvious) way is especially insidious; the expected inequality may

hold on some such interval for each value of a, but there may be no such

interval on which the inequality holds for all values of a.

From the next section onward, and all through the book, we frequently

encounter integrals that must be estimated. That is to say, be shown to be

small in absolute value, or at least not too big. So some remarks on this

topic are in order at this point.

Let f : I → C be a continuous function on some interval I ⊆ R. Consider

the problem of bounding

I

f(x) dx

from above. The inequality

I

f(x) dx ≤

I

|f(x)| dx

is basic here, but we will briefly describe some techniques that go a little

further. Note that

I

f(x) dx ≤

I

|f(x)| dx ≤

I

M dx = M(I)

if |f(x)| ≤ M for x ∈ I. Here (I) denotes the length of I. An extension of

this argument yields

I

f(x)g(x) dx ≤ M

I

|g(x)| dx

if |f(x)| ≤ M for x ∈ I. For integrals where the integrand is the product

of two functions, one of which is oscillatory, integration by parts is often

useful. We have

b

a

f(x)g(x) dx = f(b)G(b) −

b

a

f (x)G(x) dx

assuming f continuously differentiable on [a, b] and putting

G(x) =

x

a

g(u) du.

If g(x) oscillates on [a, b], there is a good possibility that G(x) will grow

slowly on the interval. If in addition f(x) changes fairly slowly on [a, b], its

derivative f (x) will be small in magnitude, and integration by parts may

yield a very favorable estimate of the integral of f(x)g(x) over [a, b]. This

is a standard technique for estimating Fourier transforms.