Take the 2-minute tour ×  Let $a$ be an integer, $p$ a prime
(much) greater than $a$, and $\chi$ a Dirichlet character. There
is an abundant literature on the sums $$S(\chi,a)=\sum_{i=1}^a
\chi(i),$$ called short (or incomplete) character sums, and
their average in various sense. Yet I have not found in this
literature an answer to the following specific question (perhaps
because of its very abundance):  Let us form the average of the
modulus of $S(\chi,a)$ over $p$; that is, let's define:
$$AS(p,a) = \frac{1}{p-1} \sum_{\chi} |S(\chi,a)|,$$ where the
sum is over the $p-1$ Dirichlet characters mod $p$. An easy
upper bound for $AS(p,a)$, namely $$AS(p,a) \leq \sqrt{a}$$ can
be obtained by applying Cauchy-Schwarz.  Is this upper bound
essentially the best possible? Precisely, is it true or false
that $AS(p,a)=o(\sqrt{a}),$ uniformly in $p >a$? (And what if
one restricts to the domain where $p$ is not too large, say $p <
a^K$ for some fixed constant $K$ -- so that the character sums
are not too short?)  I have d