TITLE: Any way to prove Prime Number Theorem using Hyperbolic Geometry?
QUESTION [7 upvotes]: The prime number theorem says that the density of prime numbers is inverse as the number of digits of $n$:
$$\displaystyle \frac{\{1 \leq k \leq n : \text{ prime } \}}{n} \approx \frac{1}{\log n}$$
Strategies to prove this result usually resort to estimating generating functions of various types.  And it seems to be proven over and over again, e.g. in these notes on  complex-analytic multiplicative number theory.
my issue with the above proofs is they are vary analytic and have estimates that I am not comfortable with.

In fact, instead of the prime number theorem let's try a much simpler estimate.  Let $\Lambda(x)$ be the Van Mangoldt function.  
$$ \Lambda(n) = 
\begin{cases}
\log p & \text{ if }n = p^k \\
0 & \text{otherwise}
\end{cases} = -\sum_{d|n} \mu(d) \log(d)$$
Logarithms and fractions naturally come up with discussions in the hyperbolic metric.  Is it possible to lift  classical proofs into hyperbolic geometry?  In particular, I could try the result:
$$ \sum_{n \leq x}  \Lambda(x) = x + o(x)$$
Does estimates like these of the Van Mangoldt function have a geometric interpretation, e.g. in terms of geodesics?
There seems to be already be a literature on the analogy between prime numbers and  prime geodesics in $\mathbb{H}$.  nLab seems to credit this to Sarnak and Selberg.  
$$\pi_\Gamma(x)  = \# \{ \gamma \in SL(2,\mathbb{Z}): N(\gamma) = e^{\ell(\gamma)} \leq x  \} = \int_0^x \frac{dt}{\log t} + \text{ error }  $$
In this case, there is no analogue of the Van Mangoldt function.  These results involve very difficult spectral arguments and I am worse off than I started.

REPLY [4 votes]: It is not at all clear what you have in mind.  True, there is an analogy between primes and closed geodesics on $\mathcal H/SL(2,\mathbb Z)$.  But the analogy arises a posteriori when one knows they have similar distributions. The geodesics have no particular connection to prime numbers.  Instead, lengths of geodesics are logarithms of fundamental units in real quadratic fields.  Their multiplicities class numbers of orders.
The generating function for the lengths of closed geodesics is the Selberg zeta function.  Its zeros are parametrized by eigenvalues of the Laplacian; the corresponding eigenvectors are called Maass wave forms.  This correspondence means that the analog of the Riemann Hypothesis for the Selberg zeta function is true.  If one could prove the prime number theorem by an approach such as you suggest, one would expect to get optimal bounds on the error term, as good as the Riemann hypothesis would give.  This seems unlikely to me.