Computations for Katz' Observations
Computations of Gaussian Primes
This document presents some computations on the "cumulative
distribution" function of normalized spacings between angles
of Gaussian primes, as defined by N. Katz in a lecture
given at Princeton on October 12, 1994. For my own notes
on the substance of this lecture, see
Notes on N. Katz' Lecture
Here we present some plots computed using PARI. The code
may be found here:
gaussprime.gp
Below is a description of the routines defined above:
-
listangles(m,n)
- This outputs a vector of all the angles of gaussian primes
of norm between m and n (up to symmetry)
sorted into increasing order.
-
spacings(m,n)
- This outputs a vector of all the normalized spacings for
norm between m and n (up to symmetry)
sorted into increasing order.
-
compar(m,n)
- This computes the cumulative distribution and compares the
values with a conjectural formula
-
plotcompar(m,n)
- This plots the functions (on screen) computed in
compar(m,n)
Note that in PARI, to prepare a PostScript file from the currently
viewed picture one gives the command postdraw([0,0,0]).
The PostScript plots displayed below (as GIFs) were slightly edited
thereafter.
Plot for m=1, n=100,000
The thick curve is the cumulative distribution function of the spacings
for the gaussian primes, while the thin curve is the possible limiting
exponential curve.
Plot for m=1, n=500,000
Plot for m=1, n=1,000,000
Note that convergence is particularly bad for small and large
t.
Plot for m=1,000,000, n=1,100,000
Convergence improves if we take a large range of large primes.
Other sequences
Out of general curiosity, we also tried the calculation
of the cumulative distribution of spacings for
nx - floor(nx) for irrational values of x.
The behavior is considerably different; apparently, the
cumulative distribution commonly resembles a step function.
The code for this case is
irratangle.gp
A sample plot is given below for x=sqrt(3)
and m=100,000, n=110,000
Similar pictures appear for other values of x (such as
cube root of 3 or Pi). I have found no value of x where
anything like an exponential function appears. I think it may be possible
to use the theory of continued fractions to prove the above
distribution function is in fact a step function.
Last modified: Tue Nov 25 20:14:20 CST 2003