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Finalist: Ane Espeseth og Torstein Vik

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 di erent 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 speci c 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 collection 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. 

 

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