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
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
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.