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The Riemann hypothesis says that every Nontrivial zero of the Riemann zeta function of $\zeta$ has real part $1/2$. The Lindelöf hypothesis says that, for every $\epsilon >0$,The exponent may be arbitrarily small; the implied constant may depend on that exponent. The next part proves the implication from the first hypothesis to the second.

The subpower zeta bound to the right of the critical line gives an arbitrarily small exponent at $1/2+\delta$. The functional equation gives exponent $\delta +\epsilon$ at $1/2-\delta$. The Phragmén–Lindelöf principle gives exponent $\delta /2+\epsilon$ at the midpoint. Letting these fixed positive parameters be arbitrarily small proves the Lindelöf hypothesis.