# Representation of e in terms of primes

1. Jan 30, 2014

### cryptist

We can represent π, in terms of primes by using Euler's product form of Riemann Zeta.
For example ζ(2)=(π^2)/6= ∏ p^2/(p^2-1).

Likewise, is there a representation of e that is obtained by using only prime numbers?

2. Feb 4, 2014

### cryptist

I guess there is no such known representation?

3. Feb 26, 2017

### DaTario

There are two expressions relating e to the prime number distribution:

$$e = \lim_{n\to \infty} n^{\pi(n)/n}$$

and

$$e = \lim_{n\to \infty} (p_n \#)^{1/p_n}$$

where $\pi(n)$ is the prime counting function, $p_n$ is the n-th prime and $p_n \#$ is the primorial of $p_n$. (see https://en.wikipedia.org/wiki/List_of_representations_of_e )

Best wishes,

DaTario