The contour diagram of the Riemann zeta function shown on the previous page showed both a few of its trivial zeroes on the real axis, and the first few of its non-trivial zeroes, on the line z = x + (1/2)i, where the Riemann hypothesis claims all its non-trivial zeroes should be.
One of the most basic relationships between the Riemann zeta function and the prime numbers has to do with the relative proportion of integers that are prime as one looks at larger and larger numbers. Roughly, this proportion declines as the natural logarithm of the number one is looking at.
However, the Prime Number Theorem gives a closer approximation to the distribution of primes. It states:
pi(x) = Li(x) + O(x ln(x))
where the function pi(x) is the number of primes less than or equal to x, and Li(x) is the logarithmic integral, defined as:
_ x
/ \ 1
Li(x) = / -------, du
\_/ ln(u)
2
The Prime Number theorem is equivalent to the statement that there are no non-trivial zeroes of the Riemann zeta function on the lines z = x and z = x + i.
From a symmetry property of the Riemann zeta function, it was known that if the Riemann zeta function did have any zeroes that didn't lie on the line z = x + (1/2)i, they would occur in pairs of the form z = x + yi and z = x + (1-y)i. It was also known that y could not be greater than one, and so the Prime Number Theorem narrowed matters down slightly, not allowing any zeroes on the very edges of that area.
If the Riemann hypothesis were true, a stronger form of the Prime Number Theorem would also be true - and vice versa, the two statements being equivalent:
pi(x) = Li(x) + O(sqrt(x) ln(x))
The function O(n) means "On the order of n". But n can't just be a number, it needs to be a function, for the precise definition of O(n) to be applied. Where a term such as O(f(x)) appears in an equation, it means a quantity which, for all x greater than one finite value, is less than or equal, in absolute magnitude, to another finite value times f(x).
So it doesn't matter if the other quantity being compared to O(f(x)) is consistently around 100 times f(x), or if f(x) becomes zero for a few small values of x, or if either the other quantity or the function f is sometimes negative.
So the Prime Number Theorem states that the difference between pi(x) and Li(x) cannot ultimately grow more quickly than x times ln(x), and the Riemann Hypothesis says the same for the smaller bound sqrt(x) times ln(x).