What is the Integral Representation of Pi(x)?

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Discussion Overview

The discussion revolves around the integral representation of the prime counting function Pi(x), exploring various mathematical formulations and their implications. Participants examine the accuracy and computational feasibility of different approaches to calculating Pi(x), including the use of integral transforms and asymptotic approximations.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents an integral representation of Pi(x) involving a series of transformations and integrals, suggesting that it can be computed with a specified accuracy.
  • Another participant argues that the commonly used approximation Pi(x) = x/ln(x) is not a true value but merely an asymptotic bound, which fails to provide accurate calculations for large x.
  • Concerns are raised about the computational difficulty of evaluating the proposed triple integral and whether it is more efficient than existing methods for calculating Pi(x).
  • Some participants discuss the implications of the Riemann Hypothesis on the proposed method, particularly regarding the location of zeros and their residues.
  • There is a suggestion that the exactness of the proposed expression does not necessarily translate to practical computational advantages.
  • Participants express varying opinions on the necessity of knowing the locations of zeros of the Riemann function for accurate calculations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the effectiveness or accuracy of the proposed integral representation compared to other methods. There are multiple competing views regarding the validity of the approximation Pi(x) = x/ln(x) and the computational feasibility of the discussed integral.

Contextual Notes

Limitations include unresolved mathematical steps in the proposed integral representation and the dependence on the accuracy of approximations for logarithmic functions in calculating Pi(x). The discussion also highlights the complexity of numerical analysis in evaluating integrals over infinite domains.

  • #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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