Anders Björner: A simplicial complex in number theory

Let ∆_n be the simplicial complex of squarefree positive integers less than or equal to n ordered by divisibility. It is known that the asymptotic rate of growth of its Euler characteristic is closely related to deep properties of the prime number system, such as the Prime Number Theorem and the Riemann Hypothesis.

The talk will be about the asymptotic growth behaviour of the individual Betti numbers β_k(∆_n) and of their sum. We show that ∆_n has the homotopy type of a wedge of spheres and that

∑_k β_k(∆_n) = 2n/π² + O(√n)

and, for fixed k

β_k(∆_n) ∼ n/2log n · (log log n)^k/k!

The talk will be quite general and elementary, assuming no specialized background.