Rademacher.27s contour Hardy–Littlewood circle method

ford circles: circle rests upon each fraction in lowest terms. darker circles shown fractions 0/1, 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5 , 4/5. each circle tangential base line , neighboring circles (see tangent lines circles). fractions same denominator have circles of same size.

in special case when circle method applied find coefficients of modular form of negative weight, hans rademacher found modification of contour makes series arising circle method converge exact result. describe contour, convenient replace unit circle upper half plane, making substitution z = exp(2πiτ), contour integral becomes integral τ = i τ = 1 + i. (the number replaced number on upper half plane, convenient choice.) rademacher s contour (more or less) given boundaries of ford circles 0 1, shown in diagram. replacement of line 1 + i boundaries of these circles non-trivial limiting process, can justified modular forms have negative weight, , more care can justified non-constant terms case of weight 0 (in other words modular functions).