# Normalization of spacings

• I
Gold Member
In the Wiki article on Montgomery's pair correlation conjecture https://en.wikipedia.org/wiki/Montgomery's_pair_correlation_conjecture, it is stated that the normalized spacing between one non-trivial zero γn =½+iT of the Riemann zeta function and the next γn+1 on the critical strip Re(z)= ½ is
the non-normalized interval length L= (γn+1 - γn) times (ln(z/(2π)))/(2π)) .(*)

Also in the intro it says that it is normalized by
multiplying L times 2π /ln(T). (**)
My questions are very elementary so that I can get started on understanding this; it is not a question about the conjecture or the RH per se.
[1] I am not sure what the normalization is supposed to do: make every interval
0<normalized length <1 ?
[2] I seem to have missed something in the difference between (*) and (**):
they are multiplying by two different factors, one of which is almost the reciprocal of the other. So, why the reciprocal if they want to do the same thing, and one that is answered, why almost the reciprocal (difference of a factor of 2π in the argument of the log)?
[3] Where does this expression come from: both intuitively and mathematically? It looks vaguely like an inverse of the normal function, but not quite.
[4] In the intro the author characterizes this informally as an "average" spacing. Average of what? All spacings between zeros? I am not sure how to match up this "average" with the given formulas.

Thanks.

## Answers and Replies

Gold Member
Let me make the question simpler: when one refers to normalizing a distribution, one usually means to subtract the expected value of the population from each sample and then divide the result by the population standard deviation. OK, but how does one do this in the case of the imaginary components of the zeros on the critical strip of the Riemann zeta function if one does not know all the zeros? Does one just base it on the millions of non-trivial zeros one already knows? Thanks.