How to write this summation as a function of $x$?

Question

I have a function $$Z(x) = \sum_{n=1}^x \frac{1}{\log \left( \frac{n}{x+1} \right)}$$ Can this summation be written as an elementary function?

The first few values I have computed are

$Z(1) = \frac{-1}{\log 2} \approx -1.442 \dots$

$Z(2) = \frac{ \log (2/9)}{\log(1/3) \log(2/3)} \approx -3.376 \dots$

$Z(3) = \frac{\log (256/27)}{\log(1/4) \log(3/4)} \approx -5.640 \dots$

Answer 1

I do not think that $Z(x)$ could be expressed in any closed form even using non elementary functions.

If you generate a table for large values of $x$ and plot the result, you will notice that it is "almost" a straight line. For example, generating data from $x=1000$ to $x=10000$, using a stepsize equal to $100$, a standard linear regression would give $$Z(x)=4356.07 -9.53035 x$$ to which corresponds an $R^2=0.999339$ which is very good.

For example, for $x=5000$, the regression predicts a value $-43295.7$ for a true value $-42595.7$ which is not too bad (I hope !).