bounded number of kth powers, The case for squares, k = 2, was answered by Lagrange in 1770, who proved that every positive integer is the sum of at most four squares. The general case was proved by Hilbert in 1909, using algebraic techniques which gave no explicit bounds. An important breakthrough was the application of analytic tools to the problem by Hardy and Littlewood. These techniques are known as the circle method, and give explicit upper bounds for the function G(k), the smallest number of kth powers needed, such as Vinogradov's bound Diophantine problems Diophantine problems are concerned with integer solutions to polynomial equations: one may study the distribution of solutions, that is, counting solutions according to some measure of "size" or height. An important example is the Gauss circle problem, which asks for integers points (x y) which satisfy In geometrical terms, given a circle centered about the origin in the plane with radius r, the problem asks how many integer lattice points lie on or inside the circle. It is not hard to prove that the answer is , where as . Again, the difficult part and a great achievement of analytic number theory is obtaining specific upper bounds on the error term E(r). It was shown by Gauss that . In general, an O(r) error term would be possible with the unit circle (or, more properly, the closed unit disk) replaced by the dilates of any bounded planar region with piecewise smooth boundary. Furthermore, replacing the unit circle by the unit square, the error term for the general problem can be as large as a linear function of r. Therefore, getting an error bound of the form for some in the case of the circle is a significant improvement. The first to attain this was Sierpiński in 1906, who showed . In 1915, Hardy and Landau each showed that one does not have . Since then the goal has been to show that for each fixed there exists a real number such that . In 2000 Huxley showed that , which is the best published result. Methods of analytic number theory Dirichlet series One of the most useful tools in multiplicative number theory are Dirichlet series, which are functions of a complex variable defined by an infinite series of the form Depending on the choice of coefficients , this series may converge everywhere, nowhere, or on