  We work under the A\"{\i}d\'{e}kon-Chen conditions which ensure that the
derivative martingale in a supercritical branching random walk on the line
converges almost surely to a nondegenerate nonnegative random variable that we
denote by $Z$. It is shown that $\mathbb{E} Z\mathbf{1}_{\{Z\le x\}}=\log
x+o(\log x)$ as $x\to\infty$. Also, we provide necessary and sufficient
conditions under which $\mathbb{E} Z\mathbf{1}_{\{Z\le x\}}=\log x+{\rm
const}+o(1)$ as $x\to\infty$. This more precise asymptotics is a key tool for
proving distributional limit theorems which quantify the rate of convergence of
the derivative martingale to its limit $Z$. The methodological novelty of the
present paper is a three terms representation of a subharmonic function of at
most linear growth for a killed centered random walk of finite variance. This
yields the aforementioned asymptotics and should also be applicable to other
models.
