Dilogarithm property

Question

I'm working on dilogarithms (Don Zagier 'The dilogarithm function', Matilde Lalìn 'dilogarithm, a cool function' )and I have encountered this six term relation:

$Li_2(x)+Li_2(y)+Li_2(z)=(1/2)[Li_2(-xy/z)+Li_2(-zy/x)+Li_2(-zx/y)]$

The only reported constraint on the relation is the following:

$1/x+1/y+1/z=1$

I have tried to check the validity of the relation (in matlab) generating random numbers in according to the reported constraint and sometimes the relation works and sometimes no. Let me report some examples in which the relation fails. Using real variables:

$x=1.962715864852604$

$y=3.988549763967933$

$z=4.170416429960425$

$Li_2(x)+Li_2(y)+Li_2(z)=6.545816576099683 -10.950878065787951i $

$(1/2)[Li_2(-xy/z)+Li_2(-zy/x)+Li_2(-zx/y)]=-3.323787824989676$

Using imaginary variables:

$x=0.230314461605202 - 0.963398691391064i$

$y=1.409680812671158 + 0.469606195659127i$

$z=0.208576204208603 + 1.265739254286414i$

$Li_2(x)+Li_2(y)+Li_2(z)=1.346590009329752 + 1.672440049108072i$

$(1/2)[Li_2(-xy/z)+Li_2(-zy/x)+Li_2(-zx/y)]=1.106535934706092 - 0.383838129551910i$

I have also tried to dim the relation by myself to see if there is some hidden hypotesis, but I have found the situation quite complicated, as the matter of fact dealing with complex logarithm properties as the logarithm of the product depends on the sum of arguments and there are many casistics I'm not able to order. Can someone explain me why the relation sometimes doesn't work? or is there an available paper where the relation is demontrated... I haven't found it.

Answer 1

This is not an answer but it is probably too long for a comment.

Consider the function $$f(x,y,z)=\text{Li}_2(x)+\text{Li}_2(y)+\text{Li}_2(z)-\frac{1}{2} \left(\text{Li}_2\left(-\frac{x y}{z}\right)+\text{Li}_2\left(-\frac{x z}{y}\right)+\text{Li}_2\left(-\frac{y z}{x}\right)\right)\tag 1$$ and use the condition $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1\implies z=\frac{x y}{x y-x-y}\tag 2$$ Replace $z$ in $(1)$ to get a function $g(x,y)$.

Now, what you can do is a 3D plot of function $g^2(x,y)$ and you will notice that there are "large" regions where it is not $0$ (in fact, there are many areas where the result is a complex number.

You can do the same with $\Re\left(g(x,y)^2\right)$ and with $\Re(g(x,y))^2$ and observe the same kind of trouble.

I had a look at the paper but I did not find anything which could be of any help to us. There may be some restrictions.

I hope that some user will be of more help.

Answer 2

Thank you claude, I have found right now the same relation on the wolfram website with the further hypothesis that all the moduli of x, y and z are less than 1. With these new hypothesis the relation always works well. Anyway the relation still holds in some other cases. For example in the case of real variables if the new hypothesis is fulfilled the relation holds with LHS and the RHS reals, if is not fulfilled sometimes the relation holds with LHS and the RHS complex and sometimes fails. Maybe the arguments of x,y,and z play a role in these case. I should reach the answer from a theoretical point of view.

I will follow you advice and i will investigate the pattern of the areas where the relation holds in the case of real variables (for imaginary ones I have to think a way to visualize theresults).