The corn dole (sometimes referred to as the grain dole) was a handout by the Roman government to citizens of Rome. The number of people who received the dole varied, but research shows it varying between 150,000 to 300,000+. The dole started in 123 BC by Gaius Gracchus, and in 58 BC Publius Clodius Pulcher made it completely free. Beloch used the number of qualified male citizens who received the dole as a starting point for estimating the population of Rome. Based on this, he then estimated the number of dependent women, children, and slaves. He also estimated the number of foreigners in the city as well. This resulted in a figure of ~800,000 people inhabiting Rome.
As a verification of the above estimate, Beloch estimated the grain consumption of his computed population and compared that with estimates of the total amount of grain flowing into Rome each year. Here is a table from Gerda De Kleijn's book, The Water Supply Of Ancient Rome - City Area, Water, and Population, which shows the years of the dole and number of recipients:
TODO scan table
There are several issues with this, and especially so when attempting to reconcile the figures with Rome's water supply. For starters, the narrow window of data points is particularly troubling: Beloch's data spans 60 years, but aqueduct information spans in excess of five hundred. Superimposed on our previous graphic of Rome's water supply, it's easy to get a sense of just how small the window really is:
TODO UPDATED GRAPHIC
Another question often pondered when people discuss the population of ancient cities is what the average expectation of life was. The formula for computing the average expectation of life is pretty simple:
AverageExpectationOfLife = SumOfTotalYearsLived / TotalIndividualsRecorded