Some computations of character sums

For a Dirichlet character \(\chi\) modulo q, it is a theorem that the maximum of the character sum, \(M(\chi)\), is always bigger than a constant times \(q^{1/2}\) and it is also a theorem that the percentage of \(\chi\) modulo q such that \(M(\chi) \lt \alpha q\) goes to 0 as q goes to infinity. This is the motivation for the following data, which is related to a paper that Leo Goldmakher and I wrote.

First, here is a histogram of \(M(\chi)/q^{1/2}\) for q = 1128931, with the data split into even and odd characters.

There is not much point to showing more histograms for q prime, really. They all look about the same. But you can see an animation of histograms for increasing primes.

Below are some tables of values of statistics for the distribution, split first into even and odd characters, and then with all characters. In the tables below, for each q, the minimum and maximum of \(M(\chi)\) are listed along with some p-quantiles. The q's are a randomly chosen list of increasing primes.

Do these numbers approach a limit? (The maximum does not.)

Even characters

Odd characters

All characters