A novel approach to the Lindelöf hypothesis. Issue 1 (13th September 2019) - Fokas, A S
Abstract: Lindelöf's hypothesis, one of the most important open problems in the history of mathematics, states that for large $t$, Riemann's zeta function $ζ (1/2+it)$ is of order $O(t^{ε })$ for any $ε>0$ . 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 cannot yield an asymptotic formula for the above sum, it appears that this strategy cannot lead to the proof of Lindelöf's hypothesis. Here a completely different approach is introduced. In particular, a novel linear integral equation is derived for $|ζ (σ +it)|^2, \ 0<σ <1$ whose asymptotic analysis yields asymptotic results for a certain Riemann zeta-type double exponential sum. This sum has the same structure as the sum describing the leading asymptotics of $|ζ (σ +it)|^2$, namely it involves $m_1^{-σ -it}m_2^{-σ -it}$, but its summation limits are different than those of the sum corresponding to $|ζ (σ +it)|^2$ . The analysis of the above integral equation requires the asymptotic estimation of four different integrals denoted by $I_1, I_2, \tilde{I}_3, \tilde{I}_4$, as well as the derivation of an exact relation between certain double exponential sums. Here the latter relation is derived, and also the rigourous analysis of the first two integrals $I_1$ and $I_2$ is presented. For the remaining two integrals, formal results are only derived that suggest a possible roadmap for the derivation of rigourous asymptotic results of the above double exponential sum, as well as for other sums associated with $|ζ (σ +it)|^2$ . Additional developments suggested by the above novel approach are also discussed.