# Lindelöf hypothesis

From Encyclopedia of Mathematics

*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 |

#### Comments

#### References

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

**How to Cite This Entry:**

Lindelöf hypothesis.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Lindel%C3%B6f_hypothesis&oldid=23388

This article was adapted from an original article by S.M. Voronin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article