Instanton counting on blowup. II. K-theoretic partition function

We study Nekrasov's deformed partition function Z(epsilon(1), epsilon(2), (alpha) over right arrow; q, beta) of 5-dimensional supersymmetric Yang-Mills theory compactified on a circle. Mathematically it is the generating function of the characters of the coordinate rings of the moduli spaces of instantons on R-4. We show that it satisfies a system of functional equations, called blowup equations, whose solution is unique. As applications, we prove (a) F(epsilon(1), epsilon(2), (alpha) over right arrow; q, beta) = epsilon(1)epsilon(2) log Z(epsilon(1); epsilon(2), (alpha) over right arrow; q, beta) is regular at epsilon(1) = epsilon(2) = 0 (a part of Nekrasov's conjecture), and (b) the genus 1 parts, which are first several Taylor coefficients of F(epsilon(1), epsilon(2), (alpha) over right arrow; q, beta), are written explicitly in terms of tau = d(2) F (0, 0, (alpha) over right arrow; q, beta) /d alpha(2) in rank 2 case.