%0 Generic %1 fokas2017novel %A Fokas, Athanassios S. %D 2017 %K 2017 analysis arxiv calculus number-theory paper %T A novel approach to the Lindelöf hypothesis %U http://arxiv.org/abs/1708.06607 %X Lindelöf's hypothesis, one of the most important open problems in the history of mathematics, states that for large $t$, Riemann's zeta function $\zeta(12+it)$ 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.

@misc{fokas2017novel, abstract = {Lindel{\"o}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 Lindel\"of 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 Lindel\"of 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 Lindel\"of 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 Lindel\"of hypothesis. Hence, since the above identity is valid for all $\epsilon$, this asymptotic identity suggests the validity of Lindel\"of's hypothesis. The completion of the rigorous derivation of the above results will be presented in a companion paper.}, added-at = {2018-06-28T16:56:56.000+0200}, author = {Fokas, Athanassios S.}, biburl = {https://www.bibsonomy.org/bibtex/27a56707badd6a06dbb3475c1b8f01e3b/analyst}, description = {[1708.06607] A novel approach to the Lindel\"of hypothesis}, interhash = {4e775dfa4ce623c19cd077a2b5bed109}, intrahash = {7a56707badd6a06dbb3475c1b8f01e3b}, keywords = {2017 analysis arxiv calculus number-theory paper}, note = {cite arxiv:1708.06607Comment: 58 pages, 4 Figures}, timestamp = {2018-06-28T16:56:56.000+0200}, title = {A novel approach to the Lindel\"of hypothesis}, url = {http://arxiv.org/abs/1708.06607}, year = 2017 }