# 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