Prime density

The density of the primes near $x$ , denoted $δ (x)$ , is the proportion of integers in a small window around $x$ that are prime. For $x$ large this quantity is remarkably well approximated by

δ (x) \approx \frac{1}{\log x} .

That is, an integer chosen uniformly in a neighbourhood of $x$ is prime with probability $\sim 1 / \log x$ . The primes thin out — but slowly, since $\log x$ grows slowly.

Heuristic derivation: two estimates of $\log (n!)$

The idea (as in zetamath's video Factorials, prime numbers, and the Riemann Hypothesis) is to estimate $\log (n!)$ twice — once arithmetically using primes, once analytically using calculus — and force the answers to agree.

1. Via primes (number theory)

The exponent $v_{p} (n)$ of a prime $p$ in $n!$ is given by Legendre's formula:

v_{p} (n) = \sum_{k \geq 1} ⌊ \frac{n}{p^{k}} ⌋ .

Bounding the floors, $\frac{n}{p} - 1 \leq v_{p} (n) \leq \frac{n}{p - 1}$ , so $v_{p} (n) \approx n / p$ . Taking logarithms of $n! = \prod_{p \leq n} p^{v_{p} (n)}$ :

\log (n!) = \sum_{p \leq n} v_{p} (n) \log p \approx n \sum_{p \leq n} \frac{\log p}{p} .

2. Via calculus (analytic)

$\log (n!) = \sum_{j = 1}^{n} \log j$ , which is a Riemann sum for the integral of $\log t$ :

\log (n!) \approx \int_{1}^{n} \log t d t = n \log n - n + 1 \approx n \log n - n .

3. Equating both estimates

Since both expressions approximate the same $\log (n!)$ , they must be close:

n \sum_{p \leq n} \frac{\log p}{p} \approx n \log n - n ⟹ \sum_{p \leq n} \frac{\log p}{p} \approx \log n - 1.

4. Differentiating both sides

Viewing $x$ as a continuous variable, the left side is a step function: when $x$ increases by $1$ we either add $\log (x) / x$ (if $x$ is prime) or add nothing. Its "average slope" is therefore

δ (x) \cdot \frac{\log x}{x},

where $δ (x)$ is the proportion of the time we add a full term. The derivative of the right side is $1 / x$ . Equating: