One of Olbian dedications to Achilles Pontarches made by a retiring priest contains the priest’s gratitude towards Achilles for «the continuance of the city»: ὑπ[ὲρ] τῆς πόλεως ... διαμονῆς (IOSPE I2 140). The word διαμονή is found in Greek inscriptions more than 200 times. In the majority of cases these inscriptions date from the Roman epoch and are made in honor of Roman emperors, Senate and the people. The author presents a classification of the ὑπὲρ τῆς διαμονῆς formulas. The analysis of the διαμονή-usages in Greek inscriptions of the Roman epoch demonstrates that Greek διαμονή is a calque of the Latin aeternitas. The Olbian inscription IOSPE I2 140 is an exception: the deity is thanked only for Olbia’s «continuance», there being no mention of the Romans. Another significant specimen of this sort is the inscription КБН/CIRB 36, where the original Roman cliché is applied to Bosporan king Teiranes and queen Aelia. As the author supposes, in these cases we can observe a sort of «transmission» of «Rome’s eternity» to the periphery of the Empire, to the Greek city of Olbia or Greek Bosporan Kingdom (КБН/CIRB 36) and an example of adaptation of Roman culture by the Greeks.