Elementary and combinatorial methods for counting prime numbers

Although prime numbers are elementary objects in number theory, the first non-trivial results about their distribution in history rely on analytical methods (see 10]). It was a big surprise when Erd?s 5] and Selberg 12] discovered new proofs of the celebrated prime number theorem without the help of advanced tools from (complex) analysis. However, both approaches, which are not completely unrelated (see 8]), still make use of limits, in particular the real logarithm. In this article we shall introduce a rational logarithm without using any limit, and then derive classical results first due to Euler, Chebyshev and Mertens. Moreover, we revisit all necessary elementary results about prime numbers, sometimes proven in a more combinatorial fashion than usual.