The Riemann hypothesis: The Lindelöf hypothesis

In spite of the strong numerical evidence in favor of the Riemann hypothesis, all attempts to prove it rigorously using techniques of classical analysis have fallen far short. For example, the Hadamard zero-free region actually excludes only a small part of the critical strip near Re(s) = 1 (and hence by the functional equation near Re(s) = 0). Indeed, the hypothesis seems so far out of reach of classical techniques that even hypotheses which are strictly weaker have not yet been proven.

One of these was proposed by E. Lindelöf in 1908 and is accordingly known as the Lindelöf hypothesis. Interestingly enough, it is concerned with how large |ζ(s)| can be on the line s = 1/2. There are several equivalent ways to state this hypothesis. Considering ζ(1/2+iτ) as a function of a real variable τ, one formulation is the statement that

|ζ(1/2 + iτ)| = O(τε)

for any ε > 0 as τ → ∞. In fact, this is equivalent to the requirement

|ζ(σ + iτ)| = O(τε)

for all σ > 1/2 and ε > 0. Intuitively, both of these state that ζ(s) grows rather slowly in the critical strip as Im(s) → ∞.

The relation of the hypotheses of Riemann and Lindelöf is made clearer given the provable fact that the Lindelöf hypothesis is equivalent to the requirement that the number of zeros of ζ(s) in the region 1/2 < Re(s) < 1 and T < Im(s) < T+1, grows more slowly than log(T). Using

N(σ,T) = #({ρ ∈ Z: Re(ρ) > σ and 0 < Im(ρ) < T})

as before then, more precisely,

(N(σ,T+1) – N(σ,T))/log(T) → 0 as T → ∞

for every σ > 1/2. This is implied by the Riemann hypothesis, which requires that N(σ,T) = 0 for all T and σ > 1/2, yet it doesn’t imply the Riemann hypothesis, so it is a strictly weaker condition.

Nevertheless, even the Lindelöf hypothesis remains an open problem, demonstrating just how difficult the Riemann hypothesis must be. The best result currently known along these lines, due to Hermann Weyl, is that

|ζ(1/2 + iτ)| = O(τ1/6+ε)

for any ε > 0 as τ → ∞.

It is thought that the Lindelöf hypothesis is the strongest result of this type that can be achieved by classical methods, if indeed it can be so achieved.

There are a number of other results which could be proven if the Riemann hypothesis is assumed. Apart from better estimates on the distribution of prime numbers, most such results, like the Lindelöf hypothesis, concern either the distribution of the imaginary parts of zeros of ζ(s), or the absolute values of ζ(s) or log(ζ(s)) as Im(s) becomes large. Most such results, likewise, remain unproven without assuming the Riemann hypothesis. Of course, if any were ever disproven, the Riemann hypothesis would be also – and that hasn’t happened.

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