## Primes versus Random Sets

**Goal:**

The famous prime number theorem (see the wikipedia page) states that the number of primes
up to a bound x behaves asymptotically as x/log(x). This can be translated into saying that the n-th prime is roughly n log(n).
If one believes in the Riemann Hypothesis (see the wikipedia page), the arguably most famous open problem in mathematics, then one also has very good bounds on the "error", that is, on the discrepancy between "reality" and the asymptotic prediction.

These results about the *distribution* of primes may suggest that primes are very "regular" and that most things are known. But this is not the case.
The world of prime numbers has many mysteries and many famous (and even more not well-known) conjectures and questions about primes are open,
such as the twin prime conjecture (see the wikipedia page).

In this project you will first make a collection of known facts about the distribution of primes, as well as a collection of open (and maybe also some solved)
questions about primes.
You will then create many sets of random integers (which will be hopefully huge) sharing the same distribution properties as the set of primes.
Then you will test the conjectures you found on your random sets (as well as on the primes).
In some cases, this may give a hint if the question or conjecture you are looking at are of an "arithmetic" nature or are "just" related to the distribution properties.

**Supervisors:**
Gabor Wiese, N.N.

**Difficulty level:**
Any.

**Tools:**

- Computer Algebra System such as sagemath

**Literature:**

- Books on analytic or elementary number theory.