View source for Lindelöf hypothesis

''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''

For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058960/l0589604.png" />,

<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>

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

<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>

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" />.

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" />,

<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>

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" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Lindelöf,   "Le calcul des résidus et ses applications à la théorie des fonctions" , Gauthier-Villars  (1905)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.C. Titchmarsh,   "The theory of the Riemann zeta-function" , Oxford Univ. Press  (1951)  pp. Chapt. 13</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Ivic,   "The Riemann zeta-function" , Wiley  (1985)</TD></TR></table>