Lindelöf Hypothesis
Let 
be the least upper bound of the numbers 
such that 
is bounded as 
, where 
is the Riemann zeta function. Then the Lindelöf hypothesis states that 
is the simplest function that is zero for 
and 
for 
.
The Lindelöf hypothesis is equivalent to the hypothesis that 
(Edwards 2001, p. 186).
Backlund (1918-1919) proved that the Lindelöf hypothesis is equivalent to the statement that for every 
, the number of roots in the rectangle ![<span style=]()
{T<=I[s]<=T+1,sigma<=R[s]<=1}" src="http://mathworld.wolfram.com/images/equations/LindelofHypothesis/Inline12.gif" style="height:14px; width:185px" /> grows less rapidly than 
as 
(Edwards 2001, p. 188).
REFERENCES:
Backlund, R. "Über die Beziehung zwischen Anwachsen und Nullstellen der Zeta-Funktion." Ofversigt Finka Vetensk. Soc. 61, No. 9, 1918-1919.
Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.
Lindelöf, E. "Quelque remarques sur la croissance de la fonction 
." Bull. Sci. Math. 32, 341-356, 1908.
