It depends on who you're asking. Perspective might be the key. If you define the Roman Empire as the governmental structure based in Rome, then neither can claim to be the successor. This however wouldn't be very useful, as the capital of the empire changed on a whole number of occasions. The second perspective would be that of continuity. The empire was too big too be governed by one man so it was divided. West Rome and Byzantium both were the Roman Empire. By that definition only the emperor in Constantinople could call himself Emperor of the Romans. Only he can trace back the legitimacy of his governing to Augustus and the earlier Empire. Thirdly one could argue, that when the Western Roman Empire fell in 476 it really didn't. The title of emperor could have thought to be vacated. Some of leaders of the conquest even tried to reactivate the title of Imperator in the west after the death of Romulus Augustus. It ultimately failed. After Rome fell and the eastern empire became more Greek things changed in Europe. Roman culture and language was mixed with all thee tribal languages and Latin became a lingua franca among the different peoples. In Rome itself the catholic church usurped some of the functions of the Roman Empire. Even today the pope is pontifex maximus a religious title that can be dated back to the Roman Republic. With the spread of Christianity spread the power of the pope. As stated in the beginning the title wasn't lost. So when in 800 Charlemagne was crowned emperor in the west one could say that the old title was just picked up again. If you asked in Constantinople the basileus would not have been happy calling that barbarian the Roman Emperor. So it depends on the perspective. If you have a rather relaxed view of what the Roman Empire was and if you can live with the fact, that no dynastic or legal ties connected the HRE with the pagan or early Christian Empire then you could answer yes. I one the other would argue that the Roman Empire fell in the 5th century in the west and in 1453 in the East. The HRE was a method of the Frankish kings do legitimize their rule and to connect themselves to the ancient Roman traditions. As you might have noticed, it is difficult to answer with a clear yes or no. I hope could help you a little. If you want to dive deeper I suggest you look into translatio imperii which thought of to be the legitimization for most imperial titles.