11M06 : Zeta(s) and L(s,c)
In 1859, Riemann published a paper titled "On the number of primes less than a given magnitude". In this paper, he developed a relation between a new form of the zeta function and p (x), the number of primes less than x. This relation involves the zeroes of the zeta function when taken over the complex plane.
Amongst some very difficult mathematics, including complex integration and Fourier inversion, there is the statement, given without any proof, that all the zeroes of the zeta function lie on the critical line Re(s) = 0.5. This statement is Riemann's Hypothesis.
Riemann did not claim to have a proof of this fact, saying that it was "very probable that all the roots are real". In fact, here he is referring to the xi function, a "cousin" of zeta, but the implications are the same.
In order to understand this statement more fully, it is necessary to understand the directions Riemann takes in his paper, and the underlying work in determining the distribution of the prime numbers. The mathematics involved are well known, so I will not repeat them here.
The prime number theorem
Prime numbers are those numbers only divisible by 1 and themselves. For example, 3 is a prime number, but 4 is not. Numbers that are not prime numbers are composite numbers, excepting 1, which is the unit.
The number of primes is infinite (Euclid originally proved this, and there are several alternative proofs), but this does not tell us whether the number of primes increases, remains the same, or decreases with x.
Empirical data suggests that the density of primes decreases with x – but by how much? Gauss, by 1849, had observed that p (x) ~ Li(x), where Li(x) is defined as:
This is the Prime Number Theorem (PNT), and both Hadamard and de la Vallée Poussin proved this independently in 1896, using techniques developed by Riemann.
The next question we ask is "How accurate is this estimate?". The simple answer is "not very accurate", and Riemann set out to improve upon this result.
Riemann creates the relation:
using Fourier inversion.
This approximation is far more accurate than Gauss's, and has the pleasing feature, as proved by von Koch in 1901, that p (x) = Li(x) + O(x–1/2 (logx)2), or in other words, that the error is eventually less than x—1/2+e for large enough x and all e >0.
This mathematics as it stands is fine, and works regardless of the location of any zeta zeroes.
The analytic zeta function
The motivation behind Riemann's work comes from a simple looking function. This function is simple the sum of the positive integers taken to some power, written as:
However this is only valid for Re(s)>1, due to the non-convergence of the function for Re(s)<=1. This does not mean however that the zeta function is undefined for Re(s)<=1, just that this representation is not "good" enough to do this.
The idea of analytic continuation is to extend a function into the whole complex domain. This extension is unique.
What Riemann does is fairly inspired stuff – he uses a definition of the gamma function to create an expression involving the zeta function above, and then integrates a similar function around a complex domain, amazing finds that the two have common components, and combines the two strands to produce an expression for zeta valid everywhere, namely:
As ex grows quicker than any xs, the integral always converges, except for a pole at s=1.
This does not explain why we are looking at the zeta function in the first place. The reason Riemann chooses the zeta function is Euler's Product Formula. This quite simply states that:
The RHS is the product taken over the primes. We can see Riemann's thoughts now – the prime numbers directly influence the zeroes of the zeta equation, and vice versa.
The functional equation
Riemann's next step is to develop the function equation of zeta, and use this to define the "cousin" function xi. A functional equation describes the symmetry of a function, for example consider f(x)=x. Then f(-x)=-f(x), and this is the functional equation of f(x)=x.
The functional equation of zeta is:
This is useful – now we know that if s is a zero, then so is 1-s (both pi and the gamma function have no zeroes).
Riemann then extracts the essence of the functional equation to define the xi function as:
The functional equation of xi is:
Further work involving xi gives us our first glimpse as to why Riemann's Hypothesis might be true. We can express the xi function as:
where a2n are all real and positive. We can see how for s=1/2 +it, the RHS involves the squares of pure imaginary numbers, which are always real, and so xi is real-valued on the critical line, but beyond this we are stumped.
The critical strip
Riemann's work was revolutionary, and it was a while before mathematics made any progress with it. Riemann actually created more problems than he solved, although his work and results are very significant.
For example, what do we know about the location of the zeroes of the zeta function? Riemann finds several, all with Re(s)=1/2, and concludes that all zeroes share this property.
The zeroes could be anywhere. In 1896, Hadamard, as a step in proving the PNT, proved that all the zeroes must lie inside the critical strip, defined by 0<Re(s)<1.
A common, but incorrect, argument from this point claims that all the zeroes do lie on the critical line, as (1) the zeroes are symmetric, and (2) they all lie in the critical strip.
However, these two facts are not enough to prove that the zeroes all lie on the critical line, as they impose no definite structure as to where the zeroes are.
Alternating zeta function
With the knowledge of the critical strip, we do not need to use Riemann's cumbersome zeta function, as we can define zeta via an alternating zeta function:
This is defined for Re(s)>0, and can be related to zeta by:
This has therefore defined zeta over the critical strip.
In turn, this allows us to completely define the symmetry of the zeta zeroes.
Previously it was shown that s a zero implies and is implied by (1-s) a zero. Or,
s=a+ib a zero implies and is implied by s=(1-a)-ib a zero.
Now we can say that s=a-ib and s=(1-a)+ib are also zeroes.
Expanding n-s gives:
Consider n-a+ib, and we find that only the sign of the imaginary sin component changes, but this does not affect the convergence of the zeta function.
There has been much progress beyond Riemann's work. In 1914 Hardy proved that there are an infinite number of zeroes on the critical line, Hardy and Littlewood established that there are at least KT zeroes on the critical line (K constant, T is the height being considered), and Selberg improved this to KTlogT.
Bohr and Landau, in 1914, proved that all but an infinitesimally small number of zeroes are on the real line, and more recently, Conrey, in 1989, showed that 40% of the zeroes are real. Computing power has established Riemann's Hypothesis for a large number of zeroes.
Odlyzko and te Riele in 1985 [OTR] disproved a conjecture about the limits of Merten's function M(x). The conjecture stated that |M(x)x-1/2-e | was bounded. If this was true, then this fact proves Riemann's Hypothesis. However, Odlyzko and te Riele proved that is it not bounded using zeta zeroes and extensive computation, and the argument collapsed, as did a similar argument involving the Farey series.
Proof of Riemann's Hypothesis
The proof of Riemann's Hypothesis uses these 2 facts:
It also uses the expansion of ns.
Consider s=1/2 + e + it as a zero, and therefore so is s=1/2 - e + it.
However, each term here is less than the original, and hence this is a zero too.
However, this means that ultimately we contradict (1), and so Riemann's Hypothesis must be true.
Most of this work is from:
Riemann's Zeta Function
Dover Press 2001
Disproof of the Merten's Conjecture
A.M. Odlyzko and H.J.J. te Riele