The Riemann hypothesis: Zeros of the zeta function in the critical strip

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 × 109, have verified that every single zero in the range (for T less than about 5 × 108) lies on the line Re(ρ) = 1/2. As of 2004, 1013 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%.

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