The Persian Empire reached its peak during the time of Darius the Great, around 500 BC.

To get an estimate of what fraction of the world's people lived under the rule of the Achaemenids, you need an estimate of the world population and the population of the Achaemenid Empire.

The 44% number is apparently calculated as follows:

World population is taken from Atlas of World Population History by Colin McEvedy and Richard Jones, published in 1978. The actual estimates he provides are 50 million in 1000 BC, and 150 million in 200 BC, from which they interpolate around 112 million in 500 BC.

The population of Persia is taken from the book Geschichte des Altertums (Ancient History) Volume IV by E. Meyer, published in 1954. It says the population of Persia during the reign of Darius was around 50 million.

If you do the math, it comes to around 44.6% of the world's population.

Are these numbers reliable? Not really. World population estimates have been made by different sources, and they vary significantly. If you check the linked table, you'll see that some of the recent sources (Population Reference Bureau and the UN) don't bother to estimate in the BC's, they only start with 1 AD. But the population in 1 AD, according to 10 different sources listed there, varies between 150 million and 330 million, which is a difference of over two-fold. The older you go, the more uncertain the estimates, so 500 BC would probably be even worse than that.

The population of Persia in Darius's time has been estimated at anywhere from 10 million to 80 million from different sources. 40 - 50 million seems a common estimate, which is probably why it was picked for the calculation. But given the 8 fold range, you can see that there's a lot of disagreement among sources.

If you pick the extreme ends of the two ranges, with a Persian population of 10 million and a world population of 330 million, only 3% of the world's population lived in Persia. At the other end, with a world population of 112 million and a Persian population of 80 million, 71% of the world's population lived in the Persian Empire. That's a pretty big range to choose from, 3% to 71%.

While the 44% number is based on the commonest quoted values, that doesn't automatically make it right. When you have only 2-3 sources for world population at 500 BC, and they can't agree, how reliable can these numbers be? Similarly, when the Persian population estimate at this date varies over a 8 fold range, picking the average of the two seems like good math, but how could you assign any confidence level to it?

I would take these numbers with a large grain of salt, keeping in mind that the two biggest population centers of the ancient (and modern) worlds - China and India - are excluded, and it seems unlikely that the Persian empire accounted for almost half the people in the world, while China plus India plus the rest of the world accounted for the other half. I'd be happier just saying it was huge in terms of area and population, certainly comparable to the other biggest ancient empires the world has seen. Was it the biggest? I don't think anyone can say that with any confidence.