Title: A novel approach to the Lindelöf hypothesis,
Abstract: Lindel{ö}f's hypothesis, one of the most important open problems in the
history of mathematics, states that for large $t$, Riemann's zeta function
$\zeta(\frac{1}{2}+it)$ is of order $O(t^{\varepsilon})$ for any
$\varepsilon>0$. It is well known that for large $t$, the leading order
asymptotics of the Riemann zeta function can be expressed in terms of a
transcendental exponential sum. The usual approach to the Lindelöf hypothesis
involves the use of ingenious techniques for the estimation of this sum.
However, since such estimates can not yield an asymptotic formula for the above
sum, it appears that this approach cannot lead to the proof of the Lindelöf
hypothesis. Here, a completely different approach is introduced: the Riemann
zeta function is embedded in a classical problem in the theory of complex
analysis known as a Riemann-Hilbert problem, and then, the large
$t$-asymptotics of the associated integral equation is formally computed. This
yields two different results. First, the formal proof that a certain Riemann
zeta-type double exponential sum satisfies the asymptotic estimate of the
Lindelöf hypothesis. Second, it is formally shown that the sum of
$|\zeta(1/2+it)|^2$ and of a certain sum which depends on $\epsilon$, satisfies
for large $t$ the estimate of the Lindelöf hypothesis. Hence, since the above
identity is valid for all $\epsilon$, this asymptotic identity suggests the
validity of Lindelöf's hypothesis. The completion of the rigorous derivation
of the above results will be presented in a companion paper.