# Riemann zeta function

## 1 Definition

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

which is valid (in fact, absolutely convergent) for all complex numbers   $s$ with $\operatorname{Re}(s)>1$. We list here some of the key properties  of the zeta function  .

1. 1.

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

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

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

2. 2.
3. 3.

The zeta function satisfies the functional equation

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

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

## 2 Distribution of primes

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

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

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

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

 $\lim_{s\to 1}\zeta(s)=\prod_{p\in S}\frac{1}{1-p^{-1}}.$

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.

## 3 Zeros of the zeta function

A nontrivial zero of the Riemann zeta function is defined to be a root $\zeta(s)=0$ of the zeta function with the property that $0\leq\operatorname{Re}(s)\leq 1$. Any other zero is called trivial zero of the 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 this region. It is then an easy matter to use the functional equation (3) to find all zeros of the zeta function with real part less than 0 (it turns out they are exactly the values $-2n$, for $n$ a positive integer). However, for values of $s$ with real part between 0 and 1, the situation is quite different, since we have neither a series definition nor a functional equation to fall back upon; and indeed to this day very little is known about the behavior of the zeta function inside this critical strip  of the complex plane.

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

## References

 Title Riemann zeta function Canonical name RiemannZetaFunction Date of creation 2013-03-22 12:38:01 Last modified on 2013-03-22 12:38:01 Owner alozano (2414) Last modified by alozano (2414) Numerical id 18 Author alozano (2414) Entry type Definition Classification msc 11M06 Synonym $\zeta$ function Related topic AnalyticContinuationOfRiemannZeta Related topic DedekindZetaFunction Related topic DirichletSeries Related topic EulerProduct Related topic Complex Related topic EulerProductFormula2 Related topic HarmonicSeriesOfPrimes Defines Euler product formula Defines Riemann hypothesis