Cuasi-exact Integral equation for Pi(x)/x 2

  1. Cuasi-exact Integral equation for Pi(x)/x**2

    I thinks i have solved a cuasi exact integral equation for Pi(x)/x**2 expresing this function in terms of eigenfunctions of a symmetric kernel K(s,t)=nexp(-n**2(s-t)**2+s**2(t+s)/2**(5st)-1 form the usual integral equationfor Pi(x)

    Log(R(5s))/5s=Int(0,infinite)Pi(t)/t(t**s-1) now my doubt if it this can be helpful or if it is done yet so i am waiting for your responses matt grime and others...i do not know if my work is helpful or if is good i have done my best this time i will put the .pdf soon.
     
  2. jcsd
  3. matt grime

    matt grime 9,396
    Science Advisor
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    Until I see the proof I can't comment on the proof. I would like to know what 'causi exact' means, and what you conisder the 'usual' integral equation pi(x) satisfies. I know what li(x) is in integral terms but not pi(x). And in what sense are you using the word 'solved'? Last time it was to say that if you could calculate every prime then you could work out pi(x) from a series expanision, which is almost the antithesis of solved, to me. Given your commments on other mathematicians and journal editors I hope you'll excuse the less than flattering tone of this post.

    One further point. You are using kernels. This usually requires some restriction to be placed on the kinds of functions in the function space. Note pi(x) isn't in L^2(R), so on what space of functions is this kernel valid.
     
    Last edited: Feb 29, 2004
  4. Here is the formula for Pi(x) as an expansion by eigenfunctions of a symmetric kernel...hope it can be interesting...
     

    Attached Files:

  5. matt grime

    matt grime 9,396
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    try putting it in some other format than word which, whilst normally is just evil, is unspeajekably bad for maths.

    i looked but it makes little sense owing to the typesetting in word.

    i noticed several undefined objects and some result involving kernels on the function space of the real line that i do not recall, so please define them and prove they are vaild resp.
     
  6. here is the zip in html format.....hope you can understand now the theory.
     

    Attached Files:

  7. matt grime

    matt grime 9,396
    Science Advisor
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    that's xml/html and still doesn't work for me. convert to a ps pdf or preferably dvi file which are the standards for mathematical articles
     
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