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 grows less rapidly than as (Edwards 2001, p. 188).