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Mathematics Colloquium: "Three Ways to Study Number-Theoretic Sums"
4:10pm Neill 5W
Nigel J. E. Pitt
Abstract: We will apply the elementary method of counting lattice points (due to Gauss and Dirichlet) as a tool to estimate the average size of the divisor function, and compare this to the other classical approach via Fourier analysis (due largely to Sierpinski and Voronoi). We will then use this second method to introduce some L-functions, and see how their arithmetic structure allows us to consider sums of multiplicative functions over prime numbers, in particular their relationship to theorems of "prime number theorem type" and to conjectures such as the Riemann hypothesis. Finally, we will consider some questions about sums of non-multiplicative functions. The focus of the talk will be on the ideas behind the results rather than rigor, and proofs will be sketched or omitted completely. The talk is intended for non-specialists, including students.