Originally published at 狗和留美者不得入内. You can comment here or there.
Yesterday, I saw the following Riemann zeta function identity:
.
I took some time to try to derive it myself and to my great pleasure, I succeeded.
Eventually, I realized that it suffices to show that
and
are equal as multisets. As sets, they are both representations of the set of 
-tuples of positive integers such that the third is a multiple of the least common multiple of the first two. In the latter one, the frequency of 
is the number of 
that divides both 
and 
such that 
. In the other one, if we write as 
where 
, the condition equates to 
, which corresponds to the number of 
dividing and and such that 
and with that, 
both dividing 
, which is the frequency of via the former representation.
The coefficients 
of the Dirichlet series of the LHS of that identity can be decomposed as follows:
.
The coefficients 
of the Dirichlet series of the RHS of that identity are
.
Observe how both are equivalent in that via the multiset equivalence proved above, 
determines the same multiset of 
for both and across that, the values of the same function 
are summed. Hence the two series are equal.
