# Prime Number Theorem

## Statement of the Theorem

The Prime Number Theorem describes the asymptotic distribution of prime numbers. It states that

$$
\pi(x)\sim \frac{x}{\log x}
$$

as $x\to\infty$.

Equivalently,

$$
\lim_{x\to\infty}\frac{\pi(x)\log x}{x}=1.
$$

Thus the density of primes near a large number $x$ is approximately

$$
\frac1{\log x}.
$$

The theorem gives a precise quantitative form to the observation that primes become less frequent as numbers grow larger.

## Logarithmic Integral Form

A more accurate approximation uses the logarithmic integral

$$
\operatorname{li}(x) =
\int_2^x \frac{dt}{\log t}.
$$

The Prime Number Theorem is equivalent to

$$
\pi(x)\sim \operatorname{li}(x).
$$

Since

$$
\operatorname{li}(x) =
\frac{x}{\log x}
+
\frac{x}{(\log x)^2}
+
\frac{2x}{(\log x)^3}
+\cdots,
$$

the logarithmic integral captures secondary terms ignored by the simpler approximation $x/\log x$.

## Historical Development

The theorem was conjectured in the late eighteenth century by Legendre and Gauss after extensive numerical computations.

Gauss observed empirically that prime density behaves approximately like $1/\log x$. He proposed that the number of primes up to $x$ should be approximated by the logarithmic integral.

The theorem was finally proved independently in 1896 by

- entity["people","Jacques Hadamard","French mathematician"]
- entity["people","Charles Jean de la Vallée Poussin","Belgian mathematician"]

Their proofs used complex analysis and properties of the Riemann zeta function.

## Reformulation Using Chebyshev Functions

The Prime Number Theorem is equivalent to several asymptotic statements involving weighted prime sums.

For the Chebyshev functions

$$
\vartheta(x) =
\sum_{p\leq x}\log p
$$

and

$$
\psi(x) =
\sum_{p^k\leq x}\log p,
$$

the theorem becomes

$$
\vartheta(x)\sim x
$$

and

$$
\psi(x)\sim x.
$$

The formulation involving $\psi(x)$ is especially convenient because $\psi(x)$ connects naturally with logarithmic derivatives of the zeta function.

## Connection with the Zeta Function

The proof of the Prime Number Theorem depends on the analytic properties of the Riemann zeta function

$$
\zeta(s) =
\sum_{n=1}^{\infty}\frac1{n^s}.
$$

Euler’s product formula gives

$$
\zeta(s) =
\prod_p
\left(1-\frac1{p^s}\right)^{-1}.
$$

This identity encodes prime numbers inside an analytic function.

The key analytic fact proved by Hadamard and de la Vallée Poussin is that

$$
\zeta(s)\neq0
$$

on the line

$$
\operatorname{Re}(s)=1.
$$

The absence of zeros on this line is sufficient to derive the asymptotic behavior of primes.

## Heuristic Interpretation

Suppose the probability that a large integer $n$ is prime is approximately $1/\log n$. Then the expected number of primes up to $x$ should be roughly

$$
\int_2^x \frac{dt}{\log t}.
$$

This heuristic predicts the Prime Number Theorem.

Although probabilistic reasoning alone cannot prove the theorem, it provides useful intuition about prime density.

## Error Terms

Define the error term

$$
E(x)=\pi(x)-\operatorname{li}(x).
$$

The Prime Number Theorem states only that

$$
E(x)=o\left(\frac{x}{\log x}\right).
$$

Understanding the true size of $E(x)$ is a deep problem connected with the zeros of the zeta function.

The Riemann Hypothesis implies the strong estimate

$$
E(x)=O(\sqrt{x}\log x).
$$

Without the Riemann Hypothesis, weaker bounds are known.

## Consequences

The Prime Number Theorem has many important consequences.

### Density of Primes

The proportion of primes among integers up to $x$ satisfies

$$
\frac{\pi(x)}x\sim\frac1{\log x}.
$$

Thus the density tends to zero.

### Size of the $n$-th Prime

If $p_n$ denotes the $n$-th prime, then

$$
p_n\sim n\log n.
$$

Hence primes grow approximately like $n\log n$.

### Divergence of Reciprocal Prime Sum

Using the Prime Number Theorem, one obtains

$$
\sum_p \frac1p=\infty.
$$

However, the divergence is extremely slow:

$$
\sum_{p\leq x}\frac1p
\sim
\log\log x.
$$

## Importance

The Prime Number Theorem is one of the foundational results of analytic number theory. It demonstrates that prime numbers, despite local irregularities, obey a precise global statistical law.

The theorem also established the power of complex analytic methods in arithmetic. Much of modern number theory grows from the interaction between prime distribution and analytic properties of $L$-functions.

Questions about finer error terms, short intervals, arithmetic progressions, and zero distributions all extend the ideas introduced by the Prime Number Theorem.