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Distribution of the zeros of the zeta function

  1. Sep 17, 2015 #1
    In http://www.americanscientist.org/issues/pub/the-spectrum-of-riemannium/5, the author mentions that the function P(x) = 1-(sin(πx)/(πx))2 seems to be, assuming the Riemann Hypothesis is true, to the two-point correlations of the zeros of the Riemann zeta function. Going by https://en.wikipedia.org/wiki/Radial_distribution_function, I take that to mean that "it is a measure of the probability of finding a particle at a distance of r away from a given reference particle", whereas a "particle" here would refer to a zero. Since P(x) = 1 for non-zero integers, and if the RH is correct the real part of the non-trivial zeros = ½, this would seem to imply that if y was a non-trivial zero then there exists, for every non-zero integer n, another zero z such that y-z = ± n⋅i . But this sounds wrong. Could someone please point out my error? Thanks.
     
  2. jcsd
  3. Sep 18, 2015 #2

    mfb

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    Can you explain how you reached that conclusion?
     
  4. Sep 18, 2015 #3
    Thanks for the answer, mfb.

    For all non-zero integer n, P(n) = 1. That is (if I understand Wiki's explanation), the probability of finding two zeros at a distance of n from one another is a certainty. Since every non-trivial zero of the Riemann zeta function has the form ½ + r⋅i for some real r, then the distance between any two non-trivial roots y and z would be |y-z| = |(½ + ry⋅i) - (½ + rz⋅i)| = |s⋅i| for the integer s= ry⋅i) - rz.

    I know that there are of course better approximations to the zeta function, but my question is specifically about this conclusion, which does not appear correct to me. (Once I understand the first assertion in that Scientific American article, I can proceed to misunderstand the parts about the atomic energy levels, but those questions will be put into another rubric.)
     
  5. Sep 19, 2015 #4
    I have a partial answer: the American Scientist (sorry, above I mis-cited it as coming from Scientific American) article cited above simplified the equation; in reality, it is normalized, as explained in https://en.wikipedia.org/wiki/Montgomery's_pair_correlation_conjecture. There it is explained that the equation is really 1-(sin(πx)/(πx))2 + δ(x). Alas, the Wiki article does not explicitly explain what δ is. However, it does give an expression for δn, where γn is the nth zero, as (γn+1n) log(γn/2π)/2π; am I correct in assuming that the expression really should be P(n) =1-(sin(πγn)/(πγn))2 + δnn)?
    I am not sure whether using n as the argument of the function P (which I made up, because the Wiki article only states that the expression is equal to "the pair correlation between pairs of zeros") is correct; it is supposed to jive with the Wiki explanation that "Informally, this means that the chance of finding a zero in a very short interval of length 2πL/log(T) at a distance 2πu/log(T) from a zero 1/2+iT is about L times the expression above. [1-(sin(πx)/(πx))2 + δ(x)] (The factor 2π/log(T) is a normalization factor that can be thought of informally as the average spacing between zeros with imaginary part about T.)"
    I would be grateful for further guidance.
     
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