# Inverse of $f(x) = \frac{x}{\log(x)}$

$\pi(x)$ is defined as number of primes less than or equal to x. Gauss showed that $\pi(x) \approx \frac{x}{log(x)}$. I want to calculate the inverse of that approximation because I want to estimate how far should I sieve in order to get the number of primes that I want. Here are the steps that I took.

\begin{align}y &= \frac{x}{\log(x)}\\ x &= y \log(x) \\ e^x&=e^{y\log(x)} \\ e^x &= (e^{\log(x)})^y \\ e^x &= x^y\end{align}

I am stuck at this point. How can I calculate $y$ in terms of $x$?

• For an exact formula, use the Lambert W function, but in your case a numerical method like Newton's method will do. – Jean-Claude Arbaut Aug 6 at 20:46
• I believe you can't do it analytically. – fonini Aug 6 at 20:46

Resorting to Lambert's $W$ is a little overkill.

With $$x=y\ln y,$$ we have

$$\frac x{\log x}=\frac{y\ln y}{\ln(y\ln y)}=\frac{y\ln y}{\ln y+\ln\ln y}\approx y.$$

Given that the distribution is anyway approximate, this can be a sufficient approximation of the inverse. Or you can use it as a starting value for Newton's iterations (use few of them anyway).

• $x = y / \ln y$ I think? division not multiplication? – Claude Aug 6 at 20:53
• @Claude: the problem statement says $y=x/\log x$, then $x\approx y\log y$. – Yves Daoust Aug 6 at 20:54
• oh, I see now what you are doing – Claude Aug 6 at 20:56
• @yasar Try to plug it: set $x := y\ln y$. Then $$\frac{x}{\ln x} = \frac{y\ln y}{\ln(y\ln y))} = \frac{y\ln y}{\ln y + \ln\ln y} = \frac{y}{1+\frac{\ln\ln y}{\ln y}} = y - y\frac{\ln\ln y}{\ln y} + o\left(y\frac{\ln\ln y}{\ln y} \right) = y + o(y)$$ – Clement C. Aug 6 at 21:16
• @ClementC. Indeed, but it does work here: it is known that $n (\log n + \log \log n - 1) < p(n) < n (\log n + \log \log n)$ for $n \ge 6$. – Robert Israel Aug 7 at 4:50

To be clear about what is hinted at in the comments, $$f^{-1}(y) = -y\,W(-\tfrac{1}{y})$$ where $W$ is the Lambert-$W$ function. (You can verify this in Mathematica.) If you wanted an asymptotic expansion check the wiki article and plug one in.