Logarithmic integral

The logarithmic integral that appears in the context of the prime counting function is the special function denoted by $Li (x)$ .

The (offset) logarithmic integral is defined as:

Li (x) = \int_{2}^{x} \frac{d t}{\log t}

Approximation of $π (x)$

The function $Li (x)$ provides a much more accurate approximation to $π (x)$ than the simpler estimate $\frac{x}{\log x}$ (prime number theorem). In fact, the Prime Number Theorem implies that:

π (x) \sim Li (x)

meaning that the relative error tends to zero as $x \to \infty$ :

lim_{x \to \infty} \frac{π (x)}{Li (x)} = 1

Moreover, the difference $π (x) - Li (x)$ changes sign infinitely often, as shown by Littlewood in 1914.

Logarithmic integral

Approximation of π(x)

Approximation of $π (x)$