TITLE: short character sums averaged on the character
QUESTION [7 upvotes]: 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 done some numerical experimenta, which are not really conclusive in what sense or another. I might very well have overlooked some paper in the literature treating this question. Thanks for any idea, guess, conjecture, proof or reference...

PS: here is the proof of the trivial upper bound: $AS(p,a) \leq \frac{1}{p-1} \sqrt{(p-1) \sum_{\chi} |S(\chi,a)|^2}$ by Cauchy-Schwarz, and $\sum_\chi |S(\chi,a)|^2$ can be expanded as $\sum_{\chi,i,j} \chi(i) \overline{\chi(j)}=\sum_{i,j} \sum_\chi \chi(ij^{-1})$ where $i,j$ run from $1$ to $a$ and
$ij^{-1}$ is computed modulo $p$; then the sum over $\chi$ is $0$ unless $i=j$ which
happens $a$ times, and it this case it is $p-1$, so $\sum_\chi |S(\chi,a)|^2=(p-1)a$, hence $AS(a,p)\leq \sqrt{a}$.

REPLY [6 votes]: In his beautiful talk at MSRI today (1 May 2017), Adam Harper announced the following theorem which settles this problem.  Let $p$ be a large prime, and let $x$ be less than $p$.  Put $L=\min (x,p/x)$.  Then 
$$ 
\frac{1}{p-1} \sum_{\chi \pmod p} \Big| \sum_{n\le x} \chi(n) \Big| 
\ll \frac{\sqrt{x}}{\min (({\log\log L})^{\frac 14}, ({\log \log \log p})^{\frac 14})}. 
$$ 
One might expect that the $(\log \log \log p)^{\frac 14}$ term above can be removed with more work.  In particular, Harper's result shows that when $x=o(p)$, the $L^1$ norm is $o(x)$.  MSRI should have the video of this lecture up in a short while.