---
tag: H77NYP0C68
kind: definition
name: Logarithmic integral
msc: ["33E20", "11N05"]
defines: VP98WSTZWT
uses: [0SYR10HD4R, 04DXAFN2ES]
sources:
  - url: https://en.wikipedia.org/wiki/Logarithmic_integral_function
---

## Definition

Let $\ln$ be the [[natural logarithm|04DXAFN2ES]].
For a real number $x > 0$ with $x \ne 1$, the *logarithmic integral* is
$$\operatorname{li}(x) = \int_0^x \frac{dt}{\ln t},$$
where for $x > 1$ the integral is the [[Cauchy principal value|0SYR10HD4R]] at $t = 1$:
$$\operatorname{li}(x) = \lim_{\varepsilon \to 0^+} \left( \int_0^{1 - \varepsilon} \frac{dt}{\ln t} + \int_{1 + \varepsilon}^x \frac{dt}{\ln t} \right).$$
The *offset logarithmic integral* is $\operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2) = \int_2^x \frac{dt}{\ln t}$ for $x \ge 2$.

## Remarks

The integrand tends to $0$ as $t \to 0^+$, so there is no problem at the lower end.

Example: the substitution $t = e^u$ gives $\operatorname{li}(x) = \operatorname{Ei}(\ln x)$, where $\operatorname{Ei}$ is the [[exponential integral|EV6ZDQ0NDY]].
The function $\operatorname{li}$ has exactly one positive zero, at $x = \mu \approx 1.4513$, the Ramanujan–Soldner constant, and $\operatorname{li}(2) \approx 1.045$.

Non-example: without the principal value the integral diverges for $x > 1$, because $\frac{1}{\ln t} \sim \frac{1}{t - 1}$ near $t = 1$ and $\int \frac{dt}{\abs{t - 1}}$ diverges.

The [[prime number theorem|0EHBY5Y84N]] is equivalent to $\pi(x) \sim \operatorname{li}(x)$, where $\pi$ is the [[prime-counting function|0RQSNVTFVZ]].
Integration by parts gives $\operatorname{li}(x) \sim \frac{x}{\ln x}$ as $x \to \infty$.