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
\(\ds \) \(=\) \(\ds \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x\)

$\blacksquare$


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