literacy rate in Greco-Roman world

Vergetius actually says that "nota" are a skill that Roman recruits should have.  Some probably learned to read and write in the Army, but the army in turn may have enhanced literacy back on the areas from which soldiers were recruited. The most evidence for rural literacy we have comes from a region of Holland known as Batavia, which was a major source of recruits into the Roman Army. In short, Roman recruits joined the army, learned to read, and then taught their relatives and children, many of whom eventually joined the Roman army as well. As the Roman army recruited mostly from communities of veterans stationed along the frontier, these communities likely had higher literacy rates because of the army's continued presence than the average peasant villiage in the interior of the Roman Empire.

In Greek city states like Athens, and even in Ptolemy's Alexandria, all citizens should have been literate as they were obliged to undergo some basic education. Little could be said about the "foreigners" who in reality formed the bulk of the population of the city states.

You ask a loaded question, which scholars still fiercly contend. The general consensus is that only about 10% of the population of the Roman empire was literate circa 100 AD, but this is just a guess. Some people, basing their arguments on the Pompeii graffitoes, which suggest very humble people could not only read but also write, argue for a higher level of literacy for the populace, but even these would probably not argue a rate above 20% even for urban areas. Clearly urban areas had a higher rate than the country. Certain institutions, like the Roman army, had a very high rate of literacy, as basic reading and writing was essential to manage logistics, and soldiers left epigraphic records of their service at a rate higher than most Roman citizens. We know that many women could also read and write, as the wife of a Batavian commander sends a birthday invitation preserved in the  swamps of Vindolanda, while Perpetua was literate in Latin and Greek.

The actual rate of literacy in the ancient world remains a mystery. What can be said, is that the culture assumed that a reasonable number of people were literate. Thus the edicts of emperors and governors were posted in plain view, and inscriptions on monuments were a major form of imperial propaganda.

Have any scholars ever made any investigation into the literacy in ancient Greece and Rome?