xviii Introduction

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

d(n) 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4

Table 1. Values of the divisor function

Counting the number of divisors of positive integers leads to a diﬃcult

problem known as the Dirichlet Divisor Problem that is still unsolved today.

Denoting the number of divisors of n by d(n), Table 1 shows that these

counts fluctuate a good deal. But much more regular behavior is revealed

by averaging d(n), and in fact

1

x

n≤x

d(n) ≈ log(x) + 2γ − 1, γ = 0.5772 . . .,

with an absolute error that tends to zero as x → +∞. To determine how

fast the error tends to zero is the divisor problem of Dirichlet.

Divisors and primes are the stuff of multiplicative number theory. But

there are also interesting counting problems connected with additive ques-

tions. The eighteenth-century English algebraist Edward Waring stated that

every positive integer may be expressed as a sum of a limited number of k-th

powers of nonnegative integers, the number required depending on k only.

We shall count the number of such representations for large integers when

the number of powers allowed is suﬃciently large, finding an asymptotic for-

mula by means of the Circle Method and establishing Waring’s claim. This

was first achieved by David Hilbert by a method different from the one used

here. The proof is the most elaborate in the book, though the prerequisites

are surprisingly modest. The Circle Method is related to Fourier theory, but

involves only Fourier series with finitely many terms so convergence issues

do not arise.

The achievements of analytic number theory are not entirely limited

to approximate counts. Some of the quantities estimated are not counting

numbers, and for a few problems exact rather than approximate results have

been attained. We shall cover one such case from algebraic number theory,

that of the analytic class number formula

hK =

wK|dK|1/2

2r1(K)+r2(K)πr2(K)RK

lim

s→1

ζK(s)

ζ(s)

that expresses the number hK of ideal classes of the ring of algebraic integers

of a number field K in terms of other arithmetic data. This formula is due

to Dirichlet and Richard Dedekind.