Logarithmic integral

Summary

The logarithmic integral is $\operatorname{li}(x) = \int_0^x \frac{dt}{\ln t}$, taken as a principal value at $t = 1$. By the prime number theorem, it is asymptotic to, and an excellent approximation of, the prime-counting function $\pi(x)$. The offset version $\operatorname{Li}(x) = \int_2^x \frac{dt}{\ln t}$ avoids the singularity at $1$. It is the exponential integral composed with the logarithm: $\operatorname{li}(x) = \operatorname{Ei}(\ln x)$.