Lindelöf hypothesis

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

For any $\epsilon>0$,


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


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 $\epsilon>0$,


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


[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



[a1] A. Ivic, "The Riemann zeta-function" , Wiley (1985)
How to Cite This Entry:
Lindelöf hypothesis. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.M. Voronin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article