Auerbach, Lotka, Zipf - pioneers of power-law city-size distributions

Power-law city-size distributions are a statistical regularity researched in many countries and urban systems. In this history of science treatise we reconsider Felix Auerbach's paper published in 1913. We review his analysis and find (i) that a constant absolute concentration, as introduced by him, is equivalent to a power-law distribution with exponent ≈ 1, (ii) that Auerbach describes this equivalence, and (iii) that Auerbach also pioneered the empirical analysis of city-size distributions across countries, regions, and time periods. We further investigate his legacy as reflected in citations and find that important follow-up work, e.g. A.J. Lotka 1925 and Zipf 1949, does give proper reference to his discovery - but other does not. For example, only approximately 20% of cityrelated works citing Zipf 1949 also cite Auerbach 1913. To our best knowledge, A.J. Lotka 1925 was the first to describe the power-law rank-size rule as it is analyzed today. M. Saibante 1928, building on Auerbach and Lotka, investigated the power-law rank-size rule across countries, regions, and time periods. G.K. Zipf's achievement was to embed these findings in his monumental 1949 book. We suggest that the use of "Auerbach-Lotka-Zipf law" (or "ALZ-law") is more appropriate than "Zipf's law for cities", which also avoids confusion with Zipf's law for word frequency. We end the treatise with biographical notes on Auerbach.