##### Department of Mathematics,

University of California San Diego

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### Math 196/296 - Student Colloquium

## Audrey Terras

#### UCSD

## A new kind of zeta function: When number theorymeets graph theory

##### Abstract:

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.

Host:

### October 2, 2002

### 12:00 PM

### AP&M 2402

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