If we can’t yet say for sure that Re(s) = 1/2 for all s such that ζ(s) = 0, what *can* we say? Progress towards establishing the Riemann hypothesis could be viewed in terms of giving tighter limits on Re(s). As before, we shall let Z be the set of zeros of ζ(s) in the critical strip {s ∈ ℂ; Re(s) ≤ 1}. ρ will stand for any member of Z. It’s fairly easily shown that there is a real constant c > 0 such that if ρ ∈ Z then

Re(ρ) < 1 – c Min(1, (log |Im(ρ)|)

^{-1})

This result was proven by Hadamard, and the region in which zeros cannot lie is known as Hadamard’s zero-free region. Except for minor variations, this is still the best result which is known. Min(x,y), of course, simply means the smaller (minimum) of x and y. If ρ is close to the real axis, log |Im(ρ)| is negative, so the result follows because there are no zeros with Re(ρ) = 1. The more interesting case is when Im(ρ) is large. Then log |Im(ρ)| is also large, so its reciprocal is small, but positive. The net result is that, on the basis of this estimate, Re(ρ) theoretically could come very close to 1 the larger Im(ρ) is. In other words, the possible “scatter” of zeros around the line Re(s) = 1/2 becomes larger as Im(s) does.

In spite of that, *most* zeros must be located close to the line for large values of Im(s). It has been shown that more than 99% of zeros satisfy

|Re(ρ) – 1/2| ≤ 8 / log|Im(ρ)|

Even from this relatively weak information about ρ it is possible to make an estimate for the error term E(x) in the relation π(x) = Li(x) + E(x). Namely it can be shown that

|E(x)| ≤ C x e

^{-c √(log x)}

for some positive constants c and C. Another way to write this, using the O-notation, is

E(x) = O(x e

^{-c √(log x)})

Hence E(x)/x = O(e^{-c √(log x)})). That is enough to prove the prime number theorem, since it implies the relative error E(x)/x → 0 as x → ∞. Yet it’s a very weak estimate, since it isn’t even as strong as E(x)/x = O(e^{-c log x)})) = O(x^{-c}).

It is possible to make tests of the Riemann hypothesis by counting the number of zeros whose imaginary part is less than any given T > 0. That is, we look at the number N(T) which Riemann considered (as described earlier) that is defined by

N(T) = #({ρ∈Z: 0 < Im(ρ) < T})

It can be shown, confirming one of Riemann’s subsidiary conjectures, that

N(T) = (T/2π)log(T/2π) – T/2π + O(log T)

In fact, there are ways to compute N(T), for large T, with an error of less than 1/2. But since N(T) is an integer, the computation determines N(T) exactly. It is also possible to locate precisely all the zeros on the line Re(ρ) = 1/2 that have Im(ρ) < T. All computations done so far, which extend at least to N(T) = 1.5 × 10^{9}, have verified that every single zero in the range (for T less than about 5 × 10^{8}) lies on the line Re(ρ) = 1/2. As of 2004, 10^{13} zeros in the critical strip had been identified, and all have Re(ρ) = 1/2. This is good evidence in support of the Riemann hypothesis, though it hardly proves it.

In a slightly different direction, it is possible to make estimates of how many zeros lie *off* the line Re(ρ) = 1/2. One defines, for T > 0 and σ > 1/2

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

Then a celebrated result discovered by Harald Bohr (brother of physicist Niels Bohr) and Edmund Landau in 1914 says that for any σ > 1/2,

N(σ,T)/N(T) → 0 as T → ∞

In other words, the proportion of zeros *not* on the line Re(ρ) = 1/2 is arbitrarily small for large enough T. Very little progress in this direction has been made since the 1914 Bohr-Landau theorem.

There have been a few additional results regarding how many zeros *are* on the line. G. H. Hardy and J. E. Littlewood showed that infinitely many are. Later Hardy and Atle Selberg showed that a nonzero proportion are on the line. The best currently proven estimate is that this proportion is more than 40%.