Lindelöf's conjecture

The lindelöf conjecture is a conjecture made by Ernst Leonard Lindelöf in 1905 and is still unproven today about the order of the Riemann zeta function along the “critical straight line” . ${\displaystyle \zeta (s)}$ ${\displaystyle 1/2+\mathrm {i} t}$

The Lindelöfsche conjecture says that for and for everyone . There is also the Bachmann Landau symbol . ${\displaystyle |\zeta (1/2+\mathrm {i} t)|={\mathcal {O}}(t^{\epsilon })}$ ${\displaystyle t\to \infty }$ ${\displaystyle \epsilon >0}$ ${\displaystyle {\mathcal {O}}}$

So far only weaker statements of the form could be proven: The value determined by Johannes van der Corput and Jurjen Koksma in 1930 has been gradually improved since then, most recently by Martin Huxley . ${\displaystyle |\zeta (1/2+\mathrm {i} t)|={\mathcal {O}}(t^{k+\epsilon })}$ ${\displaystyle k=1/6=0{,}166\dots }$ ${\displaystyle k=89/570=0{,}156\dots }$

The Lindelöf hypothesis is weaker than the Riemann hypothesis : With a proof of the Riemann hypothesis the Lindelöf hypothesis would also be proven, but not the other way around.

literature

• Edward Charles Titchmarsh : The Theory of the Riemann Zeta Function . 2nd edition, Clarendon Press, Oxford 1986, ISBN 0-19-853369-1 .

Individual evidence

1. ^ E. Lindelöf: Le calcul des résidus et ses applications dans la théorie des fonctions . Gauthier-Villars, Paris 1905.
2. ↑ JG. Van der Corput, J.-F- Koksma: Sur l'ordre de grandeur de la fonction ζ (s) de Riemann dans la bande critique . Annales de la faculté des sciences de l'Université de Toulouse, 3 ^e série, Volume 22, 1930, pages 1–39, PDF file .
3. ↑ MN Huxley: Exponential Sums and the Riemann Zeta Function V . Proceedings of the London Mathematical Society 2005 90 (1): 1-41, doi : 10.1112 / S0024611504014959