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In Multiplicative Number Theory, Montgomery and Vaughan provide an identity for Riemann's zeta function. $$\zeta(s)=\frac{s}{s-1}-s\int_{1}^{\infty}\left \{ u \right \}u^{-s-1}du$$ They then state

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But it seems to me there are several steps missing. Why does that sufficies to prove the inequality? Why are they only using the $\int_{1}^{\infty}\left \{ u \right \}u^{-s-1}du$ part of the identity? How does that prove the original inequality?

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The missing step is to express this intergral with the fractional part in terms of the zeta function, using original identity: $$\int_1^\infty\{u\} u^{-\sigma- 1}du = \frac{1}{\sigma - 1} - \frac{\zeta(\sigma)}{\sigma}.$$

Then plug-in into the chain of inequalities: $$0 \le \frac{1}{\sigma - 1} - \frac{\zeta(\sigma)}{\sigma} < \frac{1}{\sigma}.$$ By re-arranging we obtain what's required. A minor point is that we requiry the first inequality in the chain to be strict, but clearly this is true.

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