3.2 Empirical value of for Riemann zeta function zeros

help

annotate

Contents Next: Table 1 Up: 3. STATISTICS OF Previous: 3.1 Solution of

With the p.d.f.

for the infinite GUE now evaluated, we can further test the GUE hypothesis by calculating

for sequences of consecutive zeros

of the Riemann zeta function, using the data of [1]. Three sets of

consecutive zeros

were analyzed, the data sets starting at zero number

,

and

respectively. The quantity

, where

with

denoting the smoothed local density of zeros at

, was calculated and a histogram was constructed for the number of values out of the

that were tested that fell into the intervals

,

. In Figure 2 the corresponding empirical values of

at the points

are plotted as a histogram and compared with the value of

for the infinite GUE. The convergence towards the GUE value as the magnitude of the imaginary part increases is evident.

For further comparison the moments

for

were calculated and compared with the empirical data according to the law of large numbers prediction

The results are contained in Table 1. Again the trend is towards convergence to the GUE value. Note in particular the four figure agreement between

and

.

omp@cecm.sfu.ca