The Riemann Hypothesis asks where the zeros of \zeta sit. The Lindelöf Hypothesis asks something that sounds completely different: how big does \zeta get as you march straight up the critical line? Written s = \tfrac12 + it, the values \zeta(\tfrac12+it) wobble up and down forever as t \to \infty. The question is only how fast those wobbles are allowed to grow.

The conjecture, made by the Finnish mathematician Ernst Lindelöf around 1908, is that they barely grow at all — slower than any positive power of t. It is weaker than the Riemann Hypothesis (RH implies it), yet after more than a century it is still open, and it is a genuinely separate problem: nobody knows how to prove Lindelöf without essentially proving RH, but nobody has shown they are equivalent either.

The statement

Read the \varepsilon carefully: the claim is not that \zeta(\tfrac12+it) is bounded (it isn't — it is unbounded), nor that it grows like some particular slow function. It is that for any exponent you name, however tiny — t^{0.01}, t^{0.0001} — the function eventually stays below that curve. A power like t^{1/4} is too big; Lindelöf says the true growth beats every such power.

The order function \mu(\sigma)

To measure growth cleanly, define for each fixed real part \sigma the Lindelöf \mu-function — the smallest exponent that bounds \zeta on the vertical line \Re(s) = \sigma:

In words, \mu(\sigma) is the order of growth of \zeta along that vertical line. A couple of endpoints are easy. To the right of the strip the Dirichlet series converges and \zeta is bounded, so \mu(\sigma) = 0 for \sigma > 1. Far to the left, the functional equation trades \sigma for 1-\sigma and (through the Gamma factor) picks up a large power of t, forcing \mu(0) = \tfrac12 and more generally \mu(\sigma) = \tfrac12 - \sigma for \sigma < 0.

In this language the whole hypothesis collapses to a single number:

The trivial ceiling: convexity gives t^{1/4}

Here is what we can prove for free. The function \mu(\sigma) is convex (a general fact about the growth of analytic functions in a strip — the Phragmén–Lindelöf principle, from the same Lindelöf). Convexity means the graph of \mu lies on or below any chord joining two of its points. Join the two known endpoints of the strip, \mu(1) = 0 and \mu(0) = \tfrac12, by a straight line:

This t^{1/4} is the convexity bound, often called the "trivial" bound because you get it with no real work — just convexity and the two endpoints. The Lindelöf Hypothesis says the truth is not \tfrac14 but 0. Everything between those two numbers is the century-long battle of subconvexity: pushing the exponent down from \tfrac14 toward 0.

Seeing the gap

Below, the horizontal axis is \sigma across the critical strip and the vertical axis is the growth exponent \mu(\sigma). The upper line is the convexity ceiling (1-\sigma)/2 that we can prove; the lower, kinked line is the Lindelöf floor \max(0,\tfrac12-\sigma) that we conjecture. On the critical line \sigma = \tfrac12 the two are as far apart as they ever get: \tfrac14 up top, 0 at the bottom. The true \mu(\sigma) lives somewhere in the shaded gap, and every subconvexity result nudges the ceiling down toward the floor.

Weyl's classical estimate \zeta(\tfrac12+it)\ll t^{1/6} puts a single point at (\tfrac12,\ \tfrac16) — already well below the \tfrac14 ceiling and a long way toward the conjectured 0.

Worked example: comparing the exponents on the line

Fix \sigma = \tfrac12 and take a genuinely large height, say t = 10^{12}. The three bounds |\zeta(\tfrac12+it)| \le C\,t^{\theta} compare like this:

The pattern is the whole story of the field: the convexity exponent \tfrac14 was the free starting point; Hardy and Littlewood, then Weyl, pushed it to \tfrac16 using exponential-sum estimates; a long chain of refinements (van der Corput, Bombieri–Iwaniec, Huxley, and in 2017 Bourgain's \tfrac{13}{84}) has crept it lower still. Every one of these is a subconvexity result — an exponent strictly below \tfrac14 — and Lindelöf is the assertion that the exponent's ultimate infimum is 0. Notice how much the size shrinks: at t=10^{12} the trivial bound allows \sim 1000, Weyl only \sim 100, and Lindelöf essentially O(1) up to a whisker.

RH implies Lindelöf

The reason RH forces Lindelöf is that pushing all the zeros onto the critical line clears the region just to the right of it, and a zero-free zone there lets you bound \log\zeta(\tfrac12+it) tightly (via a contour argument on \zeta'/\zeta). The bound you extract is exactly the sub-polynomial one: under RH, \zeta(\tfrac12+it) \ll \exp\!\big(c\,\tfrac{\log t}{\log\log t}\big), which grows slower than any t^{\varepsilon} — hence \mu(\tfrac12)=0. The implication does not run backward: a proof of Lindelöf would be a spectacular result but would not, by itself, place a single zero on the line.

The zero-density face of Lindelöf

Lindelöf wears a second, purely zero-theoretic disguise. Write N(\sigma, T) for the number of non-trivial zeros \beta + i\gamma with \beta \ge \sigma and 0 < \gamma \le T. The number of zeros in a unit-height window near height T is N(\sigma, T+1) - N(\sigma, T), and unconditionally that count is O(\log T).

So the size of \zeta on the line and the clustering of zeros to the right of it are two views of one phenomenon: \zeta can only get large where zeros bunch up, so bounding the growth is the same as forbidding the bunching. This is the bridge to zero-density theorems, which prove partial versions of "few zeros stray right of the line" and are the main unconditional route toward Lindelöf.

Why moments and Lindelöf are the same question

The most fruitful modern angle attacks the moments of zeta — the averages

A single spike in |\zeta| contributes hugely to a high moment, so controlling I_k(T) for large k is a way of controlling the largest values \zeta can take. In fact Lindelöf is equivalent to a statement about every moment:

The two lowest moments are known exactly — the second moment is I_1(T) \sim T\log T (Hardy–Littlewood) and the fourth is I_2(T) \sim \tfrac{1}{2\pi^2}\,T(\log T)^4 (Ingham) — both comfortably of size T^{1+\varepsilon}. The Keating–Snaith conjectures, born from random matrix theory, predict the exact constant in every moment (I_k(T) \sim c_k\, T(\log T)^{k^2}), and each such moment estimate that gets proved chips directly at Lindelöf. This is why the moment programme and the Lindelöf Hypothesis are, in practice, the same battlefield.