Integral Representation of Riemann Zeta Function in terms of Fractional Part
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Theorem
Let $\zeta$ be the Riemann zeta function.
Let $s \in \C$ be a complex number with real part $\sigma > 1$.
Then
- $\ds \map \zeta s = \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x$
where $\fractpart x$ denotes the fractional part of $x$.
Proof
First, we observe that:
\(\ds \sum_{n \mathop = 1}^\infty \dfrac {n - 1} {n^s}\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \dfrac n {\paren {n + 1}^s}\) | Translation of Index Variable of Summation: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac n {\paren {n + 1}^s}\) | $0$th term vanishes | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {2^s} + \dfrac 2 {3^s} + \dfrac 3 {4^s} + \cdots\) |
Rearranging terms, we get:
\(\ds \sum_{n \mathop = 1}^\infty \dfrac n {n^s} - \sum_{n \mathop = 1}^\infty \dfrac 1 {n^s}\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac n {\paren {n + 1}^s}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac 1 {n^s}\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac n {n^s} - \sum_{n \mathop = 1}^\infty \dfrac n {\paren {n + 1}^s}\) |
Therefore, we have:
\(\text {(1)}: \quad\) | \(\ds \sum_{n \mathop = 1}^\infty n^{-s}\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty n \paren {n^{-s} - \paren {n + 1}^{-s} }\) |
Next we note:
\(\ds \int_n^{n + 1} x^{-s - 1} \rd x\) | \(=\) | \(\ds \intlimits {-\dfrac 1 s x^{-s} } n {n + 1}\) | ||||||||||||
\(\text {(2)}: \quad\) | \(\ds \) | \(=\) | \(\ds \dfrac 1 s \paren {n^{-s} - \paren {n + 1}^{-s} }\) |
Therefore, we have:
\(\ds \sum_{n \mathop = 1}^\infty n^{-s}\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty n \paren {n^{-s} - \paren {n + 1}^{-s} }\) | from $\paren {1}$ above | |||||||||||
\(\ds \) | \(=\) | \(\ds s \sum_{n \mathop = 1}^\infty n \int_n^{n + 1} x^{-s - 1} \rd x\) | from $\paren {2}$ above | |||||||||||
\(\ds \) | \(=\) | \(\ds s \int_1^\infty \floor x x^{-s - 1} \rd x\) | where $\floor x$ denotes the floor function of $x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds s \int_1^\infty \paren {x - \fractpart x } x^{-s - 1} \rd x\) | Definition of Fractional Part | |||||||||||
\(\ds \) | \(=\) | \(\ds s \int_1^\infty x^{-s} \rd x - s \int_1^\infty \fractpart x x^{-s - 1} \rd x\) | Linear Combination of Complex Integrals | |||||||||||
\(\ds \) | \(=\) | \(\ds s \intlimits {\dfrac 1 {-s + 1} x^{-s + 1} } 1 {\infty} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x\) |
$\blacksquare$