Reversing Logarithmic Integral Function

1. Apr 9, 2010

cameronm

The logarithmic integral function, which is what you get by integrating 1/ln(x), is closely linked to prime numbers. It approximates the number of primes smaller than x. Heres an infinite series which describes the function:

$$li(x)=\gamma+ln(ln(x))+\sum^{\infty}_{n=1}\frac{ln(x)^n}{n*n!}$$

where $$\gamma$$ is the Euler–Mascheroni constant.

This infinite series is a continuous function and maps x to li(x) on a 1-to-1 basis.
Therefore, in theory there should be an inverse function of li, right? But I'm having difficulty finding it.

Thanks for any help guys :)

2. Apr 9, 2010

CRGreathouse

As long as you skip values below, say, 2 to avoid the singularity there's an inverse. But I don't know of any nice form for it.

If you want to calculate its value, that's simple enough to do with a series expansion, a calculation of li at that point, and a correction.

3. Apr 9, 2010

cameronm

I'm looking into this for a computer program actually, so I dont need it in mathematical notion - just something I can put into an algorithm :)

Also if its easier with a finite limit on the series then thats ok. I just cant find or work out an inverse of it :S