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.