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

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

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


Proof

We have:

\(\displaystyle \sum_{n \mathop = 1}^\infty n^{-s}\) \(=\) \(\displaystyle \sum_{n \mathop = 1}^\infty n \left({ n^{-s} - \left({n + 1}\right)^{-s} }\right)\) Abel's Lemma: Formulation 2
\(\displaystyle \) \(=\) \(\displaystyle s \sum_{n \mathop = 1}^\infty n \int_n^{n+1} x^{-s - 1} \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle s \int_1^\infty \left\lfloor{x}\right\rfloor x^{-s - 1} \rd x\) where $\left\lfloor{x}\right\rfloor$ denotes the floor function of $x$
\(\displaystyle \) \(=\) \(\displaystyle \frac s {s - 1} - s \int_1^\infty \left\{ {x}\right\} x^{-s - 1} \rd x\) where $\left\{ {x}\right\}$ denotes the fractional part of $x$

$\blacksquare$


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