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.