  We analyze Mueller's QCD dipole wave function evolution in the double
logarithm approximation (DLA). Using complex analytical methods, we show that
the distribution of dipole in the wave function (gluon multiplicity
distribution) asymptotically satisfies the Koba-Nielsen-Olesen (KNO) scaling,
with a non-trivial scaling function $f(z)$ with $z=\frac n{\bar n}$. The
scaling function decays exponentially as $2(2.55)^2ze^{-\frac{z}{0.3917}}$ at
large $z$, while its growth is log-normal as $e^{-\frac{1}{2}\ln^2 z}$ for
small-$z$. A detailed analysis of the Fourier-Laplace transform of $f(z)$,
allows for performing the inverse Fourier transform, and access the
non-asymptotic bulk-region around the peak. The bulk and asymptotic results are
shown to be in good agreement with the measured hadronic multiplicities in DIS,
as reported by the H1 collaboration at HERA in the region of large $Q^2$. A
numerical tabulation of $f(z)$ is included. Remarkably, the same scaling
function is found to emerge in the resummation of double logarithms in the
evolution of jets. Using the generating function approach, we show why this is
the case. The absence of KNO scaling in non-critical and super-renormalizable
theories is briefly discussed. We also discuss the universal character of the
entanglement entropy in the KNO scaling limit, and its measurement using the
emitted multiplicities in DIS and $e^+e^-$ annihilation.
