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.


closed as unclear what you're asking by Andrés E. Caicedo, Jeremy Rouse, John Pardon, Stopple, Andy Putman Dec 27 '14 at 0:05

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    $\begingroup$ There is very little difference (if any) between the result for the van Mangoldt function you stated and the PNT without the error term estimate. As to the main question, I have no idea (in the sense of getting a simpler than usual approach from some standard geometry, not in the sense of restating the result in fancy artificial terms shedding no light on anything whatsoever) $\endgroup$ – fedja Dec 26 '14 at 2:17
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    $\begingroup$ Not exactly hyperbolic geometry, but the Langlands-Shahidi theory of Eisenstein series gives a proof of the nonvanishing of $\zeta(s)$ on the line $\Re(s) = 1$, which is equivalent to the prime number theorem. $\endgroup$ – Peter Humphries Dec 26 '14 at 2:53
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    $\begingroup$ I think the prime geodesic theorem should be credited to Huber and Selberg. Then I believe Hejhal and Sarnak refined the error terms. While this is not what you are asking for, this paper of Parkonnen and Paulin gives an interesting application of hyperbolic geometry to number theory. $\endgroup$ – Kimball Dec 26 '14 at 10:43
  • $\begingroup$ Look at this recent result by Hee Oh and Dale Winter: arxiv.org/abs/1603.00107 $\endgroup$ – j0equ1nn Aug 20 '17 at 7:55

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.


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