**Ane Espeseth (18 år) og Torstein Vik (16 år), Ålesund
Skole: Fagerlia videregående skole**

## MOTIVIC SYMBOLS AND CLASSICAL MULTIPLICATIVE FUNCTIONS

f we take two integers (also known as whole numbers), we can add them, subtract one from the other, or multiply them. For example:

5 + 3 = 8 5 – 3 = 2 5*3 = 15

In addition to these three operations, we have another one, very useful when computing probabilities, known as a binomial coecient. A binomial coecient

will take a positive integer, for instance 5, and another integer, for instance 3,

and compute how many dierent ways you can choose 3 things out of 5 things.

A computation would look like this:

This answer tells us that if we must choose 3 students out of a group of 5 students, there are 10 possible choices.

These four operations are related by certain formulas, called *lambda-ring axioms.*Any “number system” which has four operations related to each other by these axioms is called a *lambda-ring*.

The first main discovery in our report is a method for constructing infinitely many new “number systems” like this, which are almost as easy to compute with as the integers. The method is built on a technical new idea called a *motivic symbol*.

A completely different area of mathematics is the study of prime numbers. A prime number is a positive integer that cannot be factored into two smaller positive integers. For example, 7 is a prime number, but 8 is not, since it can be factored into 2 times 4. Understanding the structure of the prime numbers is one of most dicult challenges in all of mathematics. To help in this endeavour, mathematicians have invented something called *multiplicative functions*. A typical example is the *sigma function*, which computes the sum of all factors of a given number.

Many of the most important multiplicative functions can be expressed in terms of the famous Riemann zeta function by a specic kind of formula. Such functions are called *classical *in our report.

Now we can formulate our second main discovery, which is a new and surprising connection between the theory of lambda-rings and the theory of multiplicative functions. We have proved that the c*ollection of classical multiplicative functions is a lambda-ring*, in which *the operations correspond to certain well-known number-theoretic operations on multiplicative functions.* For example, addition in this lambda-ring corresponds to an operation called Dirichlet

convolution.