; in other words, is a contour integral, integrated over the circle described traversed once anticlockwise. We would like to take directly, that is, to use the unit circle contour. In the complex analysis formulation this is problematic, since the values of may not be defined there. Singularities on unit circle The problem addressed by the circle method is to force the issue of taking , by a good understanding of the nature of the singularities f exhibits on the unit circle. The fundamental insight is the role played by the Farey sequence of rational numbers, or equivalently by the roots of unity: Here the denominator , assuming that is in lowest terms, turns out to determine the relative importance of the singular behaviour of typical near . Method The Hardy–Littlewood circle method, for the complex-analytic formulation, can then be thus expressed. The contributions to the evaluation of , as , should be treated in two ways, traditionally called major arcs and minor arcs. We divide the roots of unity into two classes, according to whether or , where is a function of that is ours to choose conveniently. The integral is divided up into integrals each on some arc of the circle that is adjacent to , of length a function of (again, at our discretion). The arcs make up the whole circle; the sum of the integrals over the major arcs is to make up (realistically, this will happen up to a manageable remainder term). The sum of the integrals over the minor arcs is to be replaced by an upper bound, smaller in order than . Discussion Stated boldly like this, it is not at all clear that this can be made to work. The insights involved are quite deep. One clear source is the theory of theta functions. Waring's problem In the context of Waring's problem, powers of theta functions are the generating functions for the sum of squares function. Their analytic behaviour is known in much more accurate detail than for the cubes, for example. It is the case, as the false-colour diagram indicates, that for a theta function the 'most important' point on the boundary circle is at ; followed by , and then the two complex cube roots of unity at 7 o'clock and 11 o'clock. After that it is the fourth roots of unity and that matter most. While nothing in this