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''Lindelöf conjecture, on the behaviour of the Riemann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l0589603.png" />-function''
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''Lindelöf conjecture, on the behaviour of the Riemann $\zeta$-function''
  
For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l0589604.png" />,
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For any $\epsilon>0$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l0589605.png" /></td> </tr></table>
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$$\varlimsup_{t\to\infty}\frac{|\zeta(1/2+it)|}{t^\epsilon}=0.$$
  
It was stated by E. Lindelöf [[#References|[1]]]. The Lindelöf conjecture is equivalent to the assertion that for a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l0589606.png" /> the number of zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l0589607.png" /> that lie in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l0589608.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l0589609.png" />. The Lindelöf conjecture is therefore a consequence of the Riemann conjecture on the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l05896010.png" /> (cf. [[Riemann hypotheses|Riemann hypotheses]]). It is known (1982) that
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It was stated by E. Lindelöf [[#References|[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|Riemann hypotheses]]). It is known (1982) that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l05896011.png" /></td> </tr></table>
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$$\varlimsup_{t\to\infty}\frac{|\zeta(1/2+it)|}{t^c}=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l05896012.png" /> is a constant such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l05896013.png" />.
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where $c$ is a constant such that $0<c<6/37$.
  
There is a generalization of the Lindelöf conjecture to Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l05896014.png" />-functions: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l05896015.png" />,
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There is a generalization of the Lindelöf conjecture to Dirichlet $L$-functions: For any $\epsilon>0$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l05896016.png" /></td> </tr></table>
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$$L\left(\frac12+t,\chi\right)=O((k|t|+1)^\epsilon),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l05896017.png" /> is the modulus of the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l05896018.png" />.
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where $k$ is the modulus of the character $\chi$.
  
 
====References====
 
====References====

Revision as of 13:04, 4 October 2014

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

References

[a1] A. Ivic, "The Riemann zeta-function" , Wiley (1985)
How to Cite This Entry:
Lindelöf hypothesis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lindel%C3%B6f_hypothesis&oldid=33481
This article was adapted from an original article by S.M. Voronin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article