# Logarithmic Integral

Wave · y = f(x, t)

`y = ∫₂^(max(x, 1)) 1/log(u) du`

[Open in the app](https://www.wavelace.com/app#p=109) · [This page](https://www.wavelace.com/presets/logarithmic-integral)

### What it draws

The curve is an integral whose upper limit is the plot's own `x`: `y = ∫₂^x du / ln u`. That is `Li(x)`, the offset logarithmic integral, and it is Gauss's estimate of how many primes there are up to `x`. It is zero at `x = 2` and climbs, more slowly than `x`, since the integrand `1/ln u` shrinks as `u` grows. At `x = 10` the integral is `5.12`, where there are four primes. At `x = 20` it is `8.86` against eight. The estimate runs a little high, and keeps doing so.

Left of 2 the integral runs backwards and is negative, `−0.92` at `x = 1.5`. At `u = 1` the logarithm is zero and `1/ln u` has a pole, so the integral down to 1 is infinite. The formula holds its upper limit at 1 below that, `max(x, 1)`, and the curve is blank for every `x ≤ 1`.

### Why 1/ln u

Primes thin out. Near a large number `u`, about one number in every `ln u` is prime: one in 4.6 near 100. Summing that density from 2 to `x` is this integral, so `Li(x)` is the count such a density predicts. The prime number theorem says the guess is right in the limit: the true count `π(x)` and `Li(x)` have a ratio that tends to 1. The simpler estimate `x/ln x` has the same limit but is worse: at a million, `Li` overshoots the 78,498 primes by about 130, and `x/ln x` falls short by more than 6,000.

### History

Carl Friedrich Gauss, at 15 in 1792 or 1793, noticed in his tables of primes that their density near `n` was about `1/ln n`; he described the estimate in a letter to Johann Encke in 1849. Adrien-Marie Legendre published the estimate `x/(ln x − 1.08366)` in 1808. The theorem behind both was proved in 1896 by Jacques Hadamard and Charles-Jean de la Vallée Poussin with Riemann's zeta function. `Li(x)` exceeds `π(x)` at every `x` ever computed, yet John Littlewood proved in 1914 that the difference changes sign infinitely often. The first crossing is known to lie below `1.4·10³¹⁶`.

### Try

- Compare the simpler estimate, `x/log(x)`: it runs under the primes where `Li` runs over, reaching only 6.7 at `x = 20`.
- Plot the integrand, `1/log(x)`: the chance that a number near `x` is prime, one in three near 20.
- Divide by `x`, `integral(1/log(u), u, 2, max(x, 1))/x`: the fraction of the numbers up to `x` expected to be prime, falling slowly, 0.44 at `x = 20`.
- Raise `Span of x` to 40: the curve keeps climbing, to 14.8 at the far edge, where the primes number twelve.

### Read more

- [Logarithmic integral function](https://en.wikipedia.org/wiki/Logarithmic_integral_function)
- [Prime-counting function](https://en.wikipedia.org/wiki/Prime-counting_function)
- [Prime number theorem](https://en.wikipedia.org/wiki/Prime_number_theorem)
- [Skewes's number](https://en.wikipedia.org/wiki/Skewes%27s_number)