# Lindelöf hypothesis

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Lindelöf conjecture, on the behaviour of the Riemann -function

For any ,

It was stated by E. Lindelöf [1]. The Lindelöf conjecture is equivalent to the assertion that for a fixed the number of zeros of that lie in the domain is . The Lindelöf conjecture is therefore a consequence of the Riemann conjecture on the zeros of (cf. Riemann hypotheses). It is known (1982) that

where is a constant such that .

There is a generalization of the Lindelöf conjecture to Dirichlet -functions: For any ,

where is the modulus of the character .

#### 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