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

We have:

\(\ds \sum_{n \mathop = 1}^\infty n^{-s}\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty n \paren {n^{-s} - \paren {n + 1}^{-s} }\) Abel's Lemma: Formulation 2
\(\ds \) \(=\) \(\ds s \sum_{n \mathop = 1}^\infty n \int_n^{n + 1} x^{-s - 1} \rd x\)
\(\ds \) \(=\) \(\ds s \int_1^\infty \floor x x^{-s - 1} \rd x\) where $\floor x$ denotes the floor function of $x$
\(\ds \) \(=\) \(\ds \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x\)

$\blacksquare$


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