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Published on February 26th, 2014 | by Emily Corbett

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The Riemann zeta function

By Mitchell Brunton, University of Melbourne

This student took part in the 2012/13 AMSI Vacation Research Scholarship program. For more information on this years program please click here

The Riemann zeta function is a famous function in mathematics: it takes an argument, let’s call it s, and outputs the sum of the reciprocals of the natural numbers, each raised to that power. So, for example, if we gave the function 2, it would output (1/1 + 1/4 + 1/9 + 1/16 + …). The multiple zeta function is a similar function, the only difference being it can take numerous inputs.

In mathematics we often take a nice system, discern its key properties, and generalise it so it applies to other areas. One example of this something called a vector space, which one can think of as a generalisation of 2-dimensional or 3-dimension space. A vector space is spanned by vectors from a basis, which can be defined as the smallest collection of vectors within the vector space, which completely describe the space. For example, in 2-dimensional space our vectors are arrows, and a basis for this space might be {{1,0), (0,1)}, that is, a vector along the x-axis 1 unit, and a vector along the y-axis 1 unit. Every point (or vector) in 2-dimensional space can be described with multiples of these two vectors. Note that a vector space whose basis has two vectors is described as 2- dimensional. If there were 3 it would be three dimensional. There can be multiple bases for a given vector space, and the number of vectors in a basis remains fixed, regardless of the basis chosen.

In my project, I looked at slightly modified vector spaces called Z-modules. These spaces are similar to vector spaces, however instead of taking any fractional multiple of the vectors in out basis, we only considered integer multiples of these vectors. And in this case, the vectors were multiple zeta values. That is, the vectors in my space were real numbers (the output of the multiple zeta functions), and my task was to find bases for their integer spans. After a hefty deal of numerical computation, we were successful in our task, however are yet to find any pattern to our results.

Supervisor: Dr Alex Ghitza

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