Logarithmic integral function ($li(x)={\int }_0^x\frac{dt}{\ln t}$, Logarithm integral)

The Cauchy principal value integral $\mathrm{li}(x)=\mathrm{PV}{\int }_0^{x}dt/\log t$ is the logarithmic integral. The offset version $\mathrm{Li}(x)={\int }_2^{x}dt/\log t$ differs by a constant and is often used with the prime-counting function. It has $\mathrm{Li}(x)\sim x/\log x$.

The logarithmic integral function, denoted as \( \mathrm{Li}(x) \), is a special function that is defined as follows: \[ \mathrm{Li}(x) = \int_2^x \frac{dt}{\log(t)} \] for \( x > 1 \). The function is often used in number theory, particularly in relation to the distribution of prime numbers.