In this section we consider the distribution of Gram intervals that contain a given number of zeros. We present a new relation between the distribution and the good/bad nature of Gram points. We also review the prediction of Odlyzko [1] based on the GUE hypothesis. The GUE hypothesis is the conjecture that the distribution of the normalized spacing between zeros of the Zeta function is asymptotically equal to the distribution of the eigenvalues of random hermitian matrices with independent normal distribution of its coefficients. Such random hermitian matrices are called the Gauss unitary ensemble (GUE). Odlyzko derived the GUE prediction by assuming that a Gram interval does not differ from any other interval of that length. From this assumption it follows that the distribution is given by the probability that an interval of length equal to the Gram interval contains exactly m zeros. Table 1 shows the counts of Gram intervals that contain m zeros, and the GUE prediction.
Table 1: Counts of Gram intervals that contain m zeros, and the GUE prediction.
While the agreement with the GUE prediction is good, we argue here that the distribution of zero counts in Gram intervals is not independent of the type of Gram interval, and that it has to depend on whether the left side Gram point of the interval is good or bad. We reach this conclusion by considering a self- consistency condition between the probability of a Gram point being good or bad, and the probability that the corresponding interval contains an even or odd number of points. We set up the notation. pg and pb represent the probabilities that a given Gram Point is good or bad respectively. For the sample of a million zeros considered by us, pg = 0.7962 and pb = 0.2038. podd|good and peven|good are the probabilities that a Gram interval contains an odd or even number of zeros respectively, given that the left Gram point is good. podd|bad and peven|bad have corresponding interpretations when the left Gram point is bad. We now derive a consistency relation between these quantities, by asking the question: what is the probability of a given Gram point being good or bad, given information about the preceding Gram point?
pg ∗ podd|good + pb ∗ peven|bad = pg ,
pg ∗ peven|good + pb ∗ podd|bad = pb. (1)
The relationship between the quantities is shown in Equation 1. A given Gram point will be good if the preceding Gram point is good, and the intervening interval contains an odd number of zeros, or if the preceding Gram point is bad, and the intervening interval contains an even number of zeros. A given Gram point will be bad if the preceding Gram point is good, and the intervening interval contains an even number of zeros, or if the preceding Gram point is bad, and the intervening interval contains an odd number of zeros. This is is shown in Equation 1.
We can go further and derive a relationship between peven|bad and peven|good.
For Equation 1 to have a non-trivial solution for pg and pb, we require that
pb ∗ peven|bad = pg ∗ peven|good. Thus, peven|bad/peven|good = pg/pb ≈ 4. The observed values are peven|bad = 0.7204, peven|good = 0.1844, giving the ration 3.9. This establishes our contention that the distribution of zero counts in Gram intervals is not independent of the type of Gram interval. While we showed that the zero counts in a Gram interval depend on the good or bad nature of the Gram point, we also believe that there is a much smaller dependence on the odd or even nature of the Gram point. However, the study of this topic has to be postponed to future work.
[1] A. Odlyzko, ”The 1020-th Zero of the Riemann Zeta Function and 175 Million of its Neighbors’”, report, http://www.dtc.umn.edu/~odlyzko/unpublished/zeta.10to20.1992.pdf, (1992)