Equivalence of Riemann Zeta Function Definitions

Theorem

The various definitions of the Riemann zeta function are equivalent.

Specifically,

1. For all $\Re(s) > 1$,

$\displaystyle \zeta(s) = \prod_p \frac 1 {1-p^{-s}}$

where $p$ ranges over the primes

2. For all $\Re(s) > 0$,

$\displaystyle \zeta(s)=\frac 1 {1-2^{1-s}} \sum_{n=0}^\infty \frac 1 {2^{n+1}} \sum_{k=0}^n (-1)^k {n \choose k} (k+1)^{-s}$

agrees with the definition 1. on $\Re(s) > 1$.

3. For all $\Re(s) > 0$,

$\displaystyle \zeta(s) = \frac s {s-1} - s \int_1^\infty \{ x\} x^{-s-1}\ dx$

where $\{x\}$ denotes the fractional part of $x$.

4. Functional equation:

For all $\Re(s) > 0$,

$\displaystyle \pi^{-s/2}\Gamma(s/2)\zeta(s) = -\frac 1 {s(1-s)} + \int_1^\infty \left[ x^{s/2-1} + x^{-(s+1)/2} \right] \omega(x)\ dx$

where $\Gamma$ is the gamma function, $\displaystyle \omega(x) = \sum_{n=1}^\infty e^{-\pi n^2 x}$ and $\zeta$ is the Riemann zeta function.

Proof

By Analytic Continuation is Unique, to check that the definitions are equivalent, we need only check that they agree on some subset of $\C$.

1.

It is shown by Euler Product that this form of the zeta function agrees with the Dirichlet series definition on $\Re(s) > 1$.

$\Box$

3.

We have:

	\(\displaystyle \)	\(\displaystyle \sum_{n\geq 1} n^{-s}\)	\(=\)	\(\displaystyle \sum_{n \geq 1} n ( n^{-s} - (n+1)^{-s} )\)	\(\displaystyle \)	By Abel's Lemma
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle s \sum_{n\geq 1}n \int_n^{n+1} x^{-s-1}\ dx\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle s\int_1^\infty \lfloor x \rfloor x^{-s-1}\ dx\)	\(\displaystyle \)	Where $\lfloor x \rfloor$ denotes the integer part of $x$.
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac{s}{s-1} - s\int_1^\infty \{x\}x^{-s-1}\ dx\)	\(\displaystyle \)	Where $\{x\}$ denotes the fractional part of $x$.

$\Box$

4.

See Functional Equation for Riemann Zeta Function.

$\Box$

Notes

There are other equivalent definitions of the Riemann zeta function; all equivalences will be posted here.