What is the Integral Representation of Pi(x)?

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The discussion revolves around the integral representation of the prime counting function Pi(x), proposing a new formulation that expresses Pi(x) through a complex integral involving the Riemann zeta function and other mathematical constructs. The author presents a series of equations, including a transformation that relates sums over primes to integrals involving derivatives of functions. They argue that their method allows for the calculation of Pi(x) with a manageable error term, contrasting it with traditional methods that require precise logarithmic calculations. The conversation also touches on the computational complexity of evaluating such integrals and the potential for this approach to extend to complex values of x. Overall, the proposed integral representation is presented as a novel and exact method for understanding Pi(x).
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If you are going to make wild claims such as no one has ever written down a funtion defined on C such that values at integers are equal to the values of pi, then you ought at least to go away and check if that is true or not. I can think of many extensions to C.

As we keep pointing out, simply writing down a transformation isn't research. Do some work with it.

You may be an undiscovered genius, and if you keep going the way you are now that is how you'll stay. - insulting those whose approval you need isn't going to win you many supporters. You don't actually appear to want to learn any mathematics, or do any mathematics. Instead you seem content to write out elementary formulae that anyone could find. How about reading the papers of Selberg, Conrey, Odlyzko, Ono. Wiles, Green, Keating, et al to see what some real maths looks like? Then perhaps you can make a value judgement on your find.
 

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