# Distribution of the zeros of the zeta function

1. Sep 17, 2015

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. Sep 18, 2015

### Staff: Mentor

Can you explain how you reached that conclusion?

3. Sep 18, 2015

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.)

4. Sep 19, 2015