Edited by Ian Wright, Wednesday 24 April 2013 at 23:00
So now I know how it's all proved. Not yet halfway through M829 and we've reached the climax. Well perhaps not, but I undersand it all in principle - I think.
In M823 we covered a few equivalent statements to the actual theorem. The prime number theorem states that for large . This can be shown to be equivalent to or where and is its integral. Oh and is Mangoldt's function where whenever and zero otherwise.
Next, an integral expression is constructed that is equal to . This expression contains the zeta function and its derivative in the form . Using the residue theorem, and knowledge of the function's poles, this can be integrated and shown, as x approaches infinity, to equal .
And the prime number theorem is proved. That only took about 4 and a half years of studying maths to understand.
Prime Number Theorem
So now I know how it's all proved. Not yet halfway through M829 and we've reached the climax. Well perhaps not, but I undersand it all in principle - I think.
In M823 we covered a few equivalent statements to the actual theorem. The prime number theorem states that for large . This can be shown to be equivalent to or where and is its integral. Oh and is Mangoldt's function where whenever and zero otherwise.
Next, an integral expression is constructed that is equal to . This expression contains the zeta function and its derivative in the form . Using the residue theorem, and knowledge of the function's poles, this can be integrated and shown, as x approaches infinity, to equal .
And the prime number theorem is proved. That only took about 4 and a half years of studying maths to understand.