What is the Integral Representation of Pi(x)?

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SUMMARY

The integral representation of Pi(x) is established through the equality Sum(p)f(p)=Int(1,∞)pi(x)f´(x), where f(x)=x^-s. This leads to the formulation Sum(p)p^-s=g(s)=-sInt(1,∞)Pi(x)x^-(s+1)dx. The discussion highlights that calculating Pi(x) requires knowing the integral with an accuracy of 0.1, and critiques the traditional approximation Pi(x)=x/ln(x) for its limitations. The proposed method offers an exact expression for Pi(x) and discusses computational challenges associated with evaluating the integral.

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  • Understanding of integral calculus and complex analysis
  • Familiarity with the Riemann zeta function and its properties
  • Knowledge of Mellin transforms and their applications
  • Basic concepts of number theory, particularly prime number distribution
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  • #31
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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