Ottomans declare themselves the new Roman Empire

What would be the effects had the ottomans declared themselves the new Roman Empire after the fall of Constantinople
 
They did. So did their predecessors, who called themselves Rūm (Rome).

No one recognised it as legitimate. the HRE viewed itself as the legitimate Roman Empire and whilst other European monarchs may have disagreed, they all agreed a non-Christian power could not be the Roman successor, Russia called itself Rome, and the Greek Orthodoxy would not recognise the Ottoman claim either.
 
Actually HRE and Ottoman Empire privately agreed between themselves that they were both successors to Roman Empire.

Ottoman conquest of Rome would've strengthen their claims and throws Christendom out of balance and race towards national churches. That wouldn't do much about European sentiment towards the empire, but it will be strong and secure enough to last and keep bothering Europeans about how they are the effing Roman Empire.
 
They did as Kaiser-I-Rum, whether they would be recognized is another matter depending on diplomacy, and even then it would be implicit. Ivan the Terrible despite being Orthodox recognized both the Holy Roman Emperor and Ottoman Sultan as his equals, although that more or less depended on his mood.
 
They did. So did their predecessors, who called themselves Rūm (Rome).

No one recognised it as legitimate. the HRE viewed itself as the legitimate Roman Empire and whilst other European monarchs may have disagreed, they all agreed a non-Christian power could not be the Roman successor, Russia called itself Rome, and the Greek Orthodoxy would not recognise the Ottoman claim either.

Yeah, the Sultanate of Rum was, literally, the Sultanate of Rome. Everyone refuses to transliterate it.
 
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