ON PRIMES, GRAPHS AND COHOMOLOGY OLIVER KNILL Abstract. The counting function on the natural numbers de- fines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or count- ing functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtration is related to the Mertens function, the Poincar´e-Hopf in- dices at critical points correspond to the values of the Moebius function. The Morse inequalities link number theoretical quantities like the prime counting functions relevant for the distribution of primes with cohomological properties of the graphs. The just given picture is a special case of a discrete Morse cohomology equivalent to simplicial cohomology. The special example considered here is a case where the graph is the Barycentric refinement of a finite simple graph. 1. Summary 1.1. For an integer n ≥ 2, let G(n) be the graph with vertex set V n = {k | 2 ≤ k ≤ n, k square free } and edges consisting of unordered pairs (a, b) in V , where either a divides b or b divides a. We see this sequence of graphs as a Morse filtration G(n)= {f ≤ n}, where f is the Morse function f (n)= n on the Barycentric refinement G of the complete graph P on the spectrum of the ring Z. The Mertens function M (n) relates by χ(G(n)) = 1 - M (n) to the Euler characteristic χ(G(n)). By Euler-Poincar´e, this allows to express M (n) cohomologically as the sum ∑ n k=1 (-1) k b k =1 - M (n) through Betti numbers b k and interpret the values -μ(k) of the M¨obius function as Poincar´e-Hopf indices i f (x)= 1 - χ(S - f (x)) of the counting function f (x)= x. The additional 1 in the Mertens-Euler relationship appears, because the integer 1 is not included in the set of critical points. The summation of the M¨ obius values illustrates a case for the Poincar´e-Hopf theorem ∑ x i f (x)= χ(G(n)) [5]. The function f is Morse in the sense that in its Morse filtration G(x)= {f ≤ x } the unit sphere S (x) of any integer x, Date : August 20, 2016. 1991 Mathematics Subject Classification. 05C10, 57M15, 68R10, 53A55, 37Dxx. Key words and phrases. Prime numbers, Morse theory, Graph Theory, Topology. 1 arXiv:1608.06877v1 [math.CO] 22 Aug 2016