# Probability density function for Riemann-zeta zeros

Curious about the expected probability distribution for the spacing between Riemann zeta zeros, of the form $$s_n=\sigma+it_n$$, where $$\sigma=0.5$$ and $$t_n$$ is the imaginary part of the $$n$$-th zero.

The mean spacing between zeros decreases slowly as the height $$t_n$$ goes up, somewhat confounding the issue, but taking a narrow slice of 100,000 zeros starting at the billionth zero ($$t_{1000000000}$$) should make that variation negligible. Here's a histogram of the spacings in that region: The longer upper tail precludes a normal distribution. The red curve is a best fit gamma distribution, which doesn't quite do it either. The mean spacing at the beginning of the range covered is $$0.351087$$ compared to $$0.351073$$ at the end, so that variation is small.

What is the closest distribution to model the spacing?

## 1 Answer

Just looking at other distributions, the three-parameter Burr type XII distribution fits pretty well. Here's the result as above around $$t_{1e9}$$: The Burr type XII pdf is:

$$f(x|\alpha,c,k)={{kc\over \alpha}({x\over \alpha})^{c-1}\over{(1+({x\over a})^c)^{k+1}}}$$

For the above fit,

$$\alpha = 0.837947, c=2.76292$$ and $$k=8.68889$$

However, the best fit parameters change at other heights, e.g. at $$t_{10e9}$$: $$\alpha = 0.776292, c=2.73405$$ and $$k=9.38153$$

Note the mean has shifted down slightly. The mean is given by:

$$\mu = \alpha k\Gamma(k-1/c)\Gamma(1+1/c)/\Gamma(k+1)$$

The Burr type XII distribution is used most often to study mortality, survival, failure rates and the like. I guess the interval between a zeta zero and the next could be thought of as its lifespan.