Riemann Zeta Function as a Multiple Integral
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Theorem
For $n \in \Z_{> 0}$, the Riemann zeta function is given by:
- $\ds \map \zeta n = \int_{\closedint 0 1^n} \frac 1 {1 - \prod_{i \mathop = 1}^n x_i} \prod_{i \mathop = 1}^n \rd x_i$
where $\closedint 0 1^n$ denotes the Cartesian $n$th power of the closed real interval $\closedint 0 1$.
Proof
\(\ds \int_{\closedint 0 1^n} \frac 1 {1 - \prod_{i \mathop = 1}^n x_i} \prod_{i \mathop = 1}^n \rd x_i\) | \(=\) | \(\ds \int_{\closedint 0 1^n} \sum_{j \mathop = 1}^\infty \paren {\prod_{i \mathop = 1}^n x_i}^{j - 1} \prod_{i \mathop = 1}^n \rd x_i\) | Sum of Infinite Geometric Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 1}^\infty \int_{\closedint 0 1^n}\prod_{i \mathop = 1}^n {x_i}^{j - 1} \prod_{i \mathop = 1}^n \rd x_i\) | Integral of Series of Positive Measurable Functions, Fubini's Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 1}^\infty \prod_{i \mathop = 1}^n \int_0^1 x^{j - 1}_i \rd x_i\) | Fubini's Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 1}^\infty \prod_{i \mathop = 1}^n \intlimits {\frac {x^j} j } 0 1\) | Integral of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 1}^\infty \prod_{i \mathop = 1}^n \frac 1 j\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 1}^\infty \frac 1 {j^n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \zeta n\) | Definition of Riemann Zeta Function |
$\blacksquare$