The world had three Roman heirs:



-the eastern roman empire (first called byzantine by German historian Hieronymus Wolf in 1557)



- the Holy Roman Empire (Imperium Romanum Sacrum Germanicæ, Heiliges Römisches Reich Deutscher Nation), in a certain way it´s a legitime heir because it was a Rome´s leader who crown Charlemagne as emperor of the romans



-The last heir was Russia Ivan III of Russia who had married Sophia Paleologue. Sophia was a niece of Constantine XI, the last Byzantine emperor



In my opinion byzantine empire was the true heir of the eastern half of the empire, but the heir of the other half was HRE



Why HRE, it was Rome choice to crown Charlemagne king of romans so it was not a self claimed title



After that , for a long time, Kaisers were crown king of the Germans, King of Italy and king of the Romans only after that process they were called emperor (Caesar,Kaiser)



Another reason its translatio imperii were a victorious nation took the title of the defeated so if Germans defeated the romans, the Germans by Translatio imperii are the successor of “Caesar” title



In my opinion Russia is not an heir



That is why tzar comes from the word caesar, before Ivan III the title of the Russian imperator was not tzar, Kaiser also comes from the word caesar



Sorry for my lame English