  The leading asymptotic behaviour as $t\to \infty$ of the celebrated Riemann
zeta function $\zeta(s), \ s = \sigma + it, \quad 0<\sigma<1, \quad t>0 , \
t\to\infty,$ can be expressed in terms of a transcendental sum. The sharp
estimation of this sum remains one of the most important open problems in
mathematics with a long and illustrious history. Lindel\"of's hypothesis states
that for $\sigma=1/2$, this sum is of order $O\left(t^\varepsilon\right)$ for
every $\varepsilon>0$. We have recently introduced a novel approach for
estimating such transcendental sums: we have first embedded the Riemann zeta
function in a certain Riemann-Hilbert problem and we have began the analysis of
the large $t$-asymptotics of the associated integral equation. The asymptotic
analysis of the resulting integral equation requires the further splitting of
the relevant interval of integration into four subintervals which are defined
in terms of the small positive numbers $\{\delta_j\}_1^4$. The rigorous
asymptotic analysis of the first two relevant integrals, $I_1$ and $I_2$, was
performed in [F]. Here, the rigorous analysis is performed of the last two
integrals, $I_3$ and $I_4$. The combination of the above results yields a proof
for the analogue of Lindel{\"o}f's hypothesis for a slight variant of the
transcendental sum characterising the large $t$-asymptotics of $|\zeta(s)|^2$,
namely for a sum which differs from the latter sum only in the occurrence of a
logarithmic term which is larger than $\frac{1}{2}\ln t$ and smaller than
$t^{\varepsilon}$. Interestingly, the parameter $\varepsilon$ in Lindel\"of's
hypothesis is explicitly defined in terms of $\delta_3$.
