# Chapter 3. Analytic Number Theory

## The Basic Series

The harmonic series is the infinite series

$$
1+\frac12+\frac13+\frac14+\frac15+\cdots.
$$

In summation notation, it is written as

$$
\sum_{n=1}^{\infty}\frac1n.
$$

This is one of the first infinite series that appears in analysis, but it is also central in number theory. Its terms become smaller and tend to zero:

$$
\lim_{n\to\infty}\frac1n=0.
$$

However, this condition alone does not guarantee that the infinite sum has a finite value. The harmonic series is the standard example showing that terms may go to zero while the series still diverges.

## Partial Sums

The $N$-th partial sum of the harmonic series is

$$
H_N=\sum_{n=1}^{N}\frac1n =
1+\frac12+\frac13+\cdots+\frac1N.
$$

The numbers $H_N$ are called the harmonic numbers. They form an increasing sequence, since each new term is positive:

$$
H_{N+1}=H_N+\frac1{N+1}>H_N.
$$

The main question is whether this increasing sequence stays bounded. If the sequence $(H_N)$ has a finite limit, then the harmonic series converges. If the partial sums grow without bound, then the harmonic series diverges.

## Divergence by Grouping

The harmonic series diverges. A classical proof groups the terms in blocks whose lengths double:

$$
1+\frac12
+
\left(\frac13+\frac14\right)
+
\left(\frac15+\frac16+\frac17+\frac18\right)
+
\cdots.
$$

In each block after the first, every term is at least as large as the last term in that block. For example,

$$
\frac13+\frac14>\frac14+\frac14=\frac12,
$$

and

$$
\frac15+\frac16+\frac17+\frac18
>
4\cdot \frac18 =
\frac12.
$$

More generally, the block

$$
\frac1{2^k+1}+\frac1{2^k+2}+\cdots+\frac1{2^{k+1}}
$$

contains $2^k$ terms, and each term is at least $1/2^{k+1}$. Hence the whole block is at least

$$
2^k\cdot \frac1{2^{k+1}}=\frac12.
$$

Therefore the partial sums repeatedly gain at least $1/2$. Since this happens infinitely many times, the partial sums cannot remain bounded. Thus

$$
\sum_{n=1}^{\infty}\frac1n
$$

diverges.

## Slow Growth

Although the harmonic series diverges, it does so very slowly. The first few harmonic numbers are

$$
H_1=1,\qquad
H_2=\frac32,\qquad
H_3=\frac{11}{6},\qquad
H_4=\frac{25}{12}.
$$

The growth is approximately logarithmic:

$$
H_N \sim \log N.
$$

More precisely, there is a constant $\gamma$, called the Euler-Mascheroni constant, such that

$$
H_N=\log N+\gamma+o(1)
$$

as $N\to\infty$.

This means that even though $H_N$ grows without bound, it grows at about the same rate as $\log N$, not at the rate of $N$, $N^2$, or any positive power of $N$.

## Integral Comparison

The logarithmic growth can be seen from the integral comparison. Since $1/x$ is decreasing on $[1,\infty)$, the sum

$$
\sum_{n=1}^{N}\frac1n
$$

is closely related to the area under the curve $y=1/x$. We have

$$
\int_1^N \frac{dx}{x}=\log N.
$$

The sum $H_N$ differs from this integral by a bounded amount. This explains why the harmonic numbers grow like $\log N$.

## Role in Number Theory

The harmonic series appears throughout analytic number theory because it measures the average size of reciprocal sums. Number theorists often study sums such as

$$
\sum_{n\leq x}\frac1n,
\qquad
\sum_{p\leq x}\frac1p,
\qquad
\sum_{n\leq x}\frac{d(n)}n,
$$

where $p$ ranges over primes and $d(n)$ denotes the number of positive divisors of $n$.

The harmonic series is especially important because it connects finite arithmetic sums with logarithmic growth. This link between summation and logarithms is one of the basic mechanisms behind analytic number theory.