We investigate the relationship between patenting activity and the population size of metropolitan areas in the United States over the last two decades (1980-2001). We find a clear superlinear effect, whereby new patents are granted disproportionately in larger urban centers, thus showing increasing returns in inventing activity with respect to population size. We characterize this relation quantitatively as a power law with an exponent larger than unity. This phenomenon is commensurate with the presence of larger numbers of inventors in larger metropolitan areas, which we find follows a quantitatively similar superlinear relationship to population, while the productivity of individual inventors stays essentially constant across metropolitan areas. We also find that structural measures of the patent co-authorship network although weakly correlated to increasing rates of patenting, are not enough to explain them. Finally, we show that R&D establishments and employment in other creative professions also follow superlinear scaling relations to metropolitan population size, albeit possibly with different exponents.