An Round-Up of the Circle Problem

An Round-Up of the Circle Problem

Mon, 18/01/2010
16:00 		Timothy Trudgian (Oxford) Junior Number Theory Seminar Add to calendar SR1
An Round-Up of the Circle Problem
How many integer-points lie in a circle of radius $ \sqrt{x} $? A poor man's approximation might be $ \pi x $, and indeed, the aim-of-the-game is to estimate
$$P(x) = \sharp\{(m, n) \in\mathbb{Z}: \;\; m^{2} + n^{2} \leq x\} -\pi x,$$
Once one gets the eye in to show that $ P(x) = O(x^{1/2}) $, the task is to graft an innings to reduce this bound as much as one can. Since the cricket-loving G. H. Hardy proved that $ P(x) = O(x^{\alpha}) $ can only possible hold when $ \alpha \geq 1/4 $ there is some room for improvement in the middle-order. In this first match of the Junior Number Theory Seminar Series, I will present a summary of results on $ P(x) $.