  Urban scaling and Zipf's law are two fundamental paradigms for the science of
cities. These laws have mostly been investigated independently and are often
perceived as disassociated matters. Here we present a large scale investigation
about the connection between these two laws using population and GDP data from
almost five thousand consistently-defined cities in 96 countries. We
empirically demonstrate that both laws are tied to each other and derive an
expression relating the urban scaling and Zipf exponents. This expression
captures the average tendency of the empirical relation between both exponents,
and simulations yield very similar results to the real data after accounting
for random variations. We find that while the vast majority of countries
exhibit increasing returns to scale of urban GDP, this effect is less
pronounced in countries with fewer small cities and more metropolises (small
Zipf exponent) than in countries with a more uneven number of small and large
cities (large Zipf exponent). Our research puts forward the idea that urban
scaling does not solely emerge from intra-city processes, as population
distribution and scaling of urban GDP are correlated to each other.
