What does the Lindelöf hypothesis imply?

Question

I recently read this article (https://viterbischool.usc.edu/news/2018/06/mathematician-m-d-solves-one-of-the-greatest-open-problems-in-the-history-of-mathematics/) about someone who may have proven the Lindelöf hypothesis which states that the Riemann zeta-function behaves on the critical line as $$\zeta(1/2 + it)=O(t^\varepsilon)$$ for any $\varepsilon > 0$. The article is very vague, so what implications a proof of the Lindelöf hypothesis would have? For example on the distribution of prime numbers or the Riemann hypothesis?

Answer 1

The Lindelöf Hypothesis does not imply the Riemann Hypothesis. But subconvexity bounds (i.e. bounds of the form $\lvert \zeta(\tfrac{1}{2} + it) \rvert \ll t^{\alpha}$ for $0 < \alpha < 1/4$) do imply some limit on how badly the Riemann Hypothesis can fail. Roughly speaking, a subconvexity bound with exponent $\alpha$ implies that the real parts of the zeroes can't be more than $\alpha$ away from $1/2$ on average.

For instance, the Lindelöf Hypothesis implies that there are at most $O(T^\epsilon)$ zeroes with real part greater than $3/4 + \delta$ (for any small, fixed $\delta$) up to height $T$. Sometimes this is stated by saying that LH implies a strong form of the Density Hypothesis. Backlund showed that LH implies that zero percent of the zeroes lie off the critical line (but still maybe infinitely many).

But precise statements cannot be made from the Lindelöf Hypothesis. In particular, consider the $L$-series associated to a half-integral weight modular form on $\text{GL}(2, \mathbb{Z})$ (which is very much like an $L$-function, in that it has a functional equation and meromorphic continuation --- but it doesn't have an Euler product). The Riemann Hypothesis is false for these $L$-series, but in many cases it is expected that the Lindelöf Hypothesis is true. Thus we shouldn't think of the Lindelöf Hypothesis as being too strongly coupled to the Riemann Hypothesis (as there are cases where LH is true and RH is false) or to the distribution of primes (which have much less meaning for $L$-series without Euler products).