Riemann zeta function
Leonhard Euler showed that when , the sum is equal to . Euler also found that since every number is the product of a unique combination of prime numbers, the zeta function can be expressed as an infinite product: By summing up each of these geometric series in parentheses, we arrive at the following identity (the Euler Product):
This gives a hint of why an analytic object like the zeta function could be related to number theoretic results.
Extending the zeta function
The most important properties of the zeta function are based on the fact that it extends to a meromorphic function on the full complex plane which is holomorphic except at , where there is a simple pole of residue 1. Let us see how this is done.
First, we wish to extend to the strip . To do this, we introduce the alternating zeta function For , we have or We may thus use the alternating zeta function to extend the zeta function.
Proposition. The series converges whenever .
Proof. We have Since for , it follows that Since , the series in question converges.
Now we can extend the zeta function.
Theorem 1. The function has a meromorphic extension to , and it is holomorphic there except at , where it has a simple pole of residue 1.
Proof. For , we have the extension For , we have by l'Hôpital's Rule, so the pole at is simple, and its residue is .
Now, in general, It follows that the Taylor series expansion of about is It follows that . Thus the residue of the pole is 1.
The next step is the functional equation: Let Then . This gives us a meromorphic continuation of to all of .
Zeroes of the Zeta Function
Using the Euler product, it is not too difficult to show that has no zeros for . Indeed, suppose this is the case; let . Then which converges. It follows that
From the functional equation it is evident that the zeta function has zeroes at , for a postive integer. These are called the trivial zeros. Since the gamma function has no zeros, it follows that these are the only zeros with real part less than 0.
In 1859, Georg Friedrich Bernhard Riemann, after whom the function is named, established the functional equation and proved that has infinitely many zeros in the strip . He conjectured that they all lie on the line . This is the famous Riemann Hypothesis, and to this day it remains one of the great unsolved problems of mathematics. Recently it has been proven that the function's first ten trillion zeros lie on the line , but proof of the Riemann hypothesis still eludes us.
In 1896, Jacque Hadamard and Charles-Jean de la Vallée Poussin independently proved that has no zeros on the line . From this they proved the prime number theorem. We prove this result here.
We first define the phi function,
Theorem 2. The function has a meromorphic continuation to with simple poles at the poles and zeros of , and with no other poles. The continuation is
Proof. It follows from the Euler product formula that for , Since converges when , the theorem statement follows.
Now we proceed to the main result.
Theorem 3. The zeta function has no zeros on the line .
Proof. We use the fact that .
Let . Then 1 is a zero of of order 1. Thus
Suppose now that and are zeros of of of order and , respectively. (Note that and may be zero.) Then
Now for real, positive , since . It follows that Since and must be nonnegative integers, it follows that . Thus . Since was arbitrary, it follows that has no zeros on the line .
- Koch, Helmut (trans. David Kramer), Number Theory: Algebraic Numbers and Functions. AMS 2000, ISBN 0-8218-2054-0.