Lindelöf hypothesis

Lindelöf conjecture, on the behaviour of the Riemann $\zeta$-function

For any $\epsilon>0$,

$$\varlimsup_{t\to\infty}\frac{|\zeta(1/2+it)|}{t^\epsilon}=0.$$

It was stated by E. Lindelöf [1]. The Lindelöf conjecture is equivalent to the assertion that for a fixed $\sigma\in(1/2,1)$ the number of zeros of $\zeta(s)$ that lie in the domain $\operatorname{Re}s>\sigma,T<\operatorname{Im}s<T+1$ is $o(\ln T)$. The Lindelöf conjecture is therefore a consequence of the Riemann conjecture on the zeros of $\zeta(s)$ (cf. Riemann hypotheses). It is known (1982) that

$$\varlimsup_{t\to\infty}\frac{|\zeta(1/2+it)|}{t^c}=0,$$

where $c$ is a constant such that $0<c<6/37$.

There is a generalization of the Lindelöf conjecture to Dirichlet $L$-functions: For any $\epsilon>0$,

$$L\left(\frac12+t,\chi\right)=O((k|t|+1)^\epsilon),$$

where $k$ is the modulus of the character $\chi$.

References

[1]	E. Lindelöf, "Le calcul des résidus et ses applications à la théorie des fonctions" , Gauthier-Villars (1905)
[2]	E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Oxford Univ. Press (1951) pp. Chapt. 13

Comments

The first reference of Voronin's article is false; nothing on the Lindelöf hypothesis is in "Le calcul des résidus et ses applications à la théorie des fonctions". And this is obvious: Lindelöf 's book publish date is 1905, and the Lindelöf 's artticle on the hypothesis is "Quelques remarques sur la croissance de la fonction zêta(s)", Bull. des sciences mathématiques, série 2, vol. 32, 1908.

References

[a1]	A. Ivic, "The Riemann zeta-function" , Wiley (1985)