The most famous zeta function is Riemanns. We will discuss its basicproperties, for example its expression as a product over primes due toEuler. Then we consider the analog for a finite connected graph X. Thatmeans we must discuss primes in X. They will be closed paths. Then it iseasy to figure out what the Iharas zeta function of X is. We use thiszeta function to obtain a graph analog of the prime number theoremcounting the number of primes of a certain length in X. When X isregular, the poles of the Ihara zeta function of X satisfy an analog ofthe Riemann hypothesis iff the graph is Ramanujan (meaning that itprovides a good communication network). We will give examples of graphsfor which the Riemann hypothesis is true and other examples for which itis false.