Riemann Zeta Function - Definition and Basic Properties

The Riemann zeta function is the most important function in analytic number theory, connecting the distribution of prime numbers to zeros of a complex analytic function.

DefinitionRiemann Zeta Function

The Riemann zeta function is defined for $\Re(s) > 1$ by:

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

By analytic continuation, $\zeta(s)$ extends to a meromorphic function on all of $\mathbb{C}$ with a single simple pole at $s=1$ with residue $1$ .

ExampleSpecial Values

$\zeta(2) = \frac{\pi^2}{6}$ (Basel problem)
$\zeta(4) = \frac{\pi^4}{90}$
$\zeta(2k) = (-1)^{k+1} \frac{B_{2k} (2\pi)^{2k}}{2(2k)!}$ for positive integers $k$ , where $B_{2k}$ are Bernoulli numbers
$\zeta(0) = -1/2$
$\zeta(-1) = -1/12$ (related to string theory and regularization)
$\zeta(-2n) = 0$ for positive integers $n$ (trivial zeros)

DefinitionEuler Product Formula

For $\Re(s) > 1$ :

\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}} = \prod_p \left(1 + \frac{1}{p^s} + \frac{1}{p^{2s}} + \frac{1}{p^{3s}} + \cdots \right)

This connects the sum over all integers to a product over primes, encoding the fundamental theorem of arithmetic.

The Euler product immediately proves infinitude of primes: if finitely many primes existed, the product would be finite, but $\zeta(s) \to \infty$ as $s \to 1^+$ .

Remark

The pole at $s=1$ has profound number-theoretic significance. The residue calculation:

\lim_{s \to 1^+} (s-1) \zeta(s) = 1

is equivalent to $\sum_{n \leq x} 1/n \sim \log x$ , the divergence of the harmonic series.

ExampleConnection to Prime Counting

Taking logarithms of the Euler product:

\log \zeta(s) = \sum_p \log\left(\frac{1}{1-p^{-s}}\right) = \sum_p \sum_{k=1}^{\infty} \frac{1}{k p^{ks}}

The main contribution comes from $k=1$ :

\log \zeta(s) \approx \sum_p \frac{1}{p^s}

This sum encodes the distribution of primes and diverges like $\log(s-1)$ as $s \to 1^+$ .

The zeta function transforms discrete arithmetic questions about primes into analytic questions about zeros, poles, and growth rates of a complex function—the founding insight of analytic number theory.