The title says it all.
I have looked in several books and on the internet and many say that the Riemann hypothesis can be used to determine the distribution of prime numbers but no one really explains how.
Any help would be appreciated.
This page on Riemann’s explicit formula seems to be the kind of thing you’re looking for, though I haven’t read it through all the way. Importantly, the formula he’s describing there doesn’t give π(x) (the prime distribution function) but rather another function Ψ(x) that increases by ln(p) every time it hits a prime p.
Thanks, the website has graphics that make it a bit clearer, but I’m still not absolutely sure I understand it yet.
The fundamental connection between the prime numbers and the zeta function is the remarkable Euler Product Formula, which shows that zeta of s is an infinite product of simple terms involving each prime number to the s power. You can easily verify this formula yourself by simple algebra. See the Wikipedia article on the Euler Product Formula.
Expanding on what JWT said, here’s the basic connection between the Riemann zeta function and the primes:
Let’s start with the basic fact, which comes up over and over again, that (1 - X) * (1 + X + X[sup]2[/sup] + X[sup]3[/sup] + …) = (1 + X + X[sup]2[/sup] + X[sup]3[/sup] + …) - (X + X[sup]2[/sup] + X[sup]3[/sup] + X[sup]4[/sup] + …) = 1. Thus, 1 + X + X[sup]2[/sup] + X[sup]3[/sup] + … = 1/(1 - X). [This sum only converges in the standard sense when |X| < 1, but nevermind that for now]
As a special case of this, for any prime p and any power s, we have that 1 + p[sup]s[/sup] + p[sup]2s[/sup] + p[sup]3s[/sup] + … = 1/(1 - p[sup]s[/sup]).
Suppose we multiplied this out over all primes. Then the left hand side would be (1 + 2[sup]s[/sup] + 2[sup]2s[/sup] + 2[sup]3s[/sup]…) * (1 + 3[sup]s[/sup] + 3[sup]2s[/sup] + 3[sup]3s[/sup] + …) * (1 + 5[sup]s[/sup] + 5[sup]2s[/sup] + 5[sup]3s[/sup] + …) * …, and the right hand side would be the reciprocal of (1 - 2[sup]s[/sup]) * (1 - 3[sup]s[/sup]) * (1 - 5[sup]s[/sup]) * … .
But the left hand side, when you expand it all out, becomes the same as 1[sup]s[/sup] + 2[sup]s[/sup] + 3[sup]s[/sup] + 4[sup]s[/sup] + …, with each positive integer appearing exactly once because it admits a unique prime factorization. Thus, we have that 1[sup]s[/sup] + 2[sup]s[/sup] + 3[sup]s[/sup] + 4[sup]s[/sup] + … = the reciprocal of (1 - 2[sup]s[/sup]) * (1 - 3[sup]s[/sup]) * (1 - 5[sup]s[/sup]) * …, where the left hand side is a sum over all positive integers and the right hand side is a product over all primes. This value is conventionally called Zeta(-s), thus defining the Zeta function. [Again, the sum and the product only converge in the standard sense when the real component of s is < -1, but there’s a natural way to extend this definition smoothly to all s except -1; that is, for all inputs to Zeta except 1]
And so, the Zeta function is rather closely connected to the primes, because it can be expressed as that simple product over all primes. In particular, Riemann used this to give an expression for the the prime-counting function in terms of the Zeta function, and various other work has built upon that.
You know, I once took a course in which we proved the prime number theorem, of course, I know and can readily derive Euler’s product formula, I’ve read Darbyshire’s book and I still don’t know the answer, really, to the question. However, you will get a better explanation from Darbyshire’s book than any other place.
An important fact which I don’t know how to prove, but which may help make the connection more concrete, is the following theorem proven by von Mangoldt: for x > 1, it turns out that e[sup]N[/sup]/2π is the least common multiple of the positive integers below* x, where N is x minus the sum of x[sup]ρ[/sup]/ρ over all zeros ρ of the Zeta function.
Of course, another way of describing the least common multiple of the positive integers below x is as the product, over each prime, of its highest power below x. [The function Ψ mentioned by MikeS is just the logarithm of this]. Thus, we have a concrete connection between the zeros of the Zeta function and the distribution of the primes. (It’s easy enough to massage Ψ into the function which just directly counts the number of primes below x, if one wants)
*: Well, there’s a slight caveat. The caveat is that I haven’t specified whether “below” means “strictly less than” or “less than or equal to”. Actually, what happens is that e[sup]N[/sup]/2π is the geometric average of the two different interpretations (though the distinction between interpretations only matters right at the jump discontinuities where x is a prime power).
(I should also clarify that this is not at all my field; I’m just trying to be some kind of intermediary from the genuine experts to non-mathematical laymen)