Riemann zeta function

1 Definition

The Riemann zeta function is defined to be the complex valuedfunction given by the series

$$\zeta (s):=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s},$$(1)

which is valid (in fact, absolutely convergent) for all complexnumbers $s$ with $\mathrm{Re}(s)>1$. We list here some of the keyproperties [1] of the zeta function.

1. For all $s$ with $\mathrm{Re}(s)>1$, the zeta function satisfies theEuler product formula

   $$\zeta (s)=\prod _p\frac{1}{1-p^{-s}},$$(2)

   where the product is taken over all positive integer primes $p$, and convergesuniformly in a neighborhood of $s$.

2. The zeta function has a meromorphic continuation to the entirecomplex plane with a simple pole at $s=1$, of residue $1$, and noother singularities.

3. The zeta function satisfies the functional equation

   $$\zeta (s)=2^s{\pi }^{s-1}\sin \frac{\pi s}{2}\mathrm{\Gamma }(1-s)\zeta (1-s),$$(3)

   for any $s\in ℂ$ (where $\mathrm{\Gamma }$ denotes the Gamma function).

2 Distribution of primes

The Euler product formula (2) given above expresses thezeta function as a product over the primes $p\in \mathbb{Z}$, andconsequently provides a link between the analytic properties of thezeta function and the distribution of primes in the integers. As thesimplest possible illustration of this link, we show how theproperties of the zeta function given above can be used to prove thatthere are infinitely many primes.

If the set $S$ of primes in $\mathbb{Z}$ were finite, then the Euler productformula

$$\zeta (s)=\prod _{p\in S}\frac{1}{1-p^{-s}}$$

would be a finite product, and consequently ${lim}_{s\to 1}\zeta (s)$would exist and would equal

$$\underset{s\to 1}{lim}\zeta (s)=\prod _{p\in S}\frac{1}{1-p^{-1}}.$$

But the existence of this limit contradicts the fact that $\zeta (s)$has a pole at $s=1$, so the set $S$ of primes cannot be finite.

A more sophisticated analysis of the zeta function along these linescan be used to prove both the analytic prime number theorem andDirichlet’s theorem on primes in arithmetic progressions. Proofs ofthe prime number theorem can be found in [2]and [5], and for proofs of Dirichlet’s theorem on primesin arithmetic progressions the reader may look in [3]and [7].

3 Zeros of the zeta function

A nontrivial zero of the Riemann zeta function is defined to bea root $\zeta (s)=0$ of the zeta function with the property that $0\le \mathrm{Re}(s)\le 1$. Any other zero is called trivial zero ofthe zeta function.

The reason behind the terminology is as follows. For complex numbers$s$ with real part greater than 1, the series definition (1)immediately shows that no zeros of the zeta function exist in thisregion. It is then an easy matter to use the functionalequation (3) to find all zeros of the zeta functionwith real part less than 0 (it turns out they are exactly the values$-2n$, for $n$ a positive integer). However, for values of $s$ withreal part between 0 and 1, the situation is quite different, since wehave neither a series definition nor a functional equation to fallback upon; and indeed to this day very little is known about thebehavior of the zeta function inside this critical strip of thecomplex plane.

It is known that the prime number theorem is equivalent to theassertion that the zeta function has no zeros $s$ with $\mathrm{Re}(s)=0$ or$\mathrm{Re}(s)=1$. The celebrated Riemann hypothesis asserts that all nontrivial zeros $s$ of the zeta function satisfy the much more precise equation $\mathrm{Re}(s)=1/2$. If true, the hypothesis would have profoundconsequences on the distribution of primes in theintegers [5].