Recently I've found the following paper, discussing and proving Selberg's symmetry formula: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2006/PAPERS/Balady.pdf My question concerns proofs of asymptotic relations shown on pages 7 and 8. Author claims to be using Mobius inversion formulas to derive these, but as far as I can tell, Mobius inversion can be applied when we have sums over divisors of fixed number.
Can Mobius inversion formula be applied here? If so, how? If not, can someone point me to a reference which would derive these results in alternate way?
Thanks in advance.
Another form of Möbius inversion is the following.
If for all positive $x$, we have that $$ G(x) = \sum_{n \leq x} F\left(\frac{x}{n}\right), $$ then it is also true that $$ F(x) = \sum_{n \leq x} \mu(n) G\left(\frac{x}{n}\right). $$ The converse is also true (or rather, the top holds for all positive $x$ iff the bottom holds for all positive $x$). This is the form that is used in your document. I am not sure why they only write down the divisor form and proceed to never use it.
There are some others that we can mention too. For instance, fix a positive integer $k$. Then $$ f(n) = \sum_{d^k \mid n} g(n/d^k) \iff g(n) = \sum_{d^k \mid n} \mu(d)f(n/d^k). $$ Also, $$ f(n) = \prod_{d \mid n} g(d) \iff g(n) = \prod_{d \mid n} f(n/d)^{\mu(d)}.$$ If you can prove ordinary Möbius inversion, you can prove these. The proofs are deceptively identical. You might interpret the last as Möbius inversion with multiplicative operations instead of additive, which suggests the true generality of Möbius inversion.