# Lindelöf Hypothesis

## Hypothesis

The Lindelöf hypothesis is a conjecture about the rate of growth of the Riemann zeta function on the critical line that is implied by the Riemann Hypothesis.

It states that:

$\forall \epsilon \in \R_{> 0}: \map \zeta {\dfrac 1 2 + i t} \text{ is } \map {\mathcal O} {t^\epsilon}$

as $t \to \infty$ (see big-O notation).

Since $\epsilon$ can always be replaced by a smaller value, we can also write the conjecture as:

$\forall \epsilon \in \R_{> 0}: \map \zeta {\dfrac 1 2 + i t} \text{ is } \map o {t^\epsilon}$

as $t \to \infty$ (see little-o notation).

## Source of Name

This entry was named for Ernst Leonard Lindelöf.