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I'm trying to understand the proof of PNT by Don Zagier. But his proof is too simplified so I can't understand it. I got stumped at step II: $\zeta(s)-1/(s-1)$ extends holomorphically to $\Re(s)>0$.

The proof is available here: http://mathdl.maa.org/images/upload_library/22/Chauvenet/Zagier.pdf

Can anyone post a detailed proof that $\zeta(s)-1/(s-1)$ extends holomorphically to $\Re(s)>0$

Or please tell me which book has a detailed proof so I can look it up?

edit:
Zagier's proof:
$\displaystyle \zeta(s)-\frac{1}{s-1} = \sum_{n=1}^{\infty} \int_{n}^{n+1} (\frac{1}{n^s}-\frac{1}{x^s}) dx$
$\displaystyle \left|\int_{n}^{n+1} (\frac{1}{n^s}-\frac{1}{x^s}) dx\right| \leq \frac{|s|}{(n^{\Re(s)+1})}$
therefore $\displaystyle \zeta(s)-\frac{1}{s-1}$ extends holomorphically to $\Re(s)>0$

My question: We have a function
(i) $f(s)=\sum_{1}^{\infty} g_n(s) \quad \forall \Re(s)>0$
(ii) $g_n(s) \leq |s|/(n^{\Re(s)+1}) \quad \forall \Re(s)>0$

Is (i) and (ii) the necessary and sufficient condition that $f(s)$ has holomorphic continuation?
Does there exist functions $f$ and $g_n$ that satisfy (i) and (ii), but does not have holomorphic continuation?
Do we need to prove other conditions, for example $g_n$ must be continuous/holomorphic?

  • 2
    What part *exactly* you don't understand in that proof? It looks to me pretty straightforward...2012-08-25
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    @alek: You might notice that I merged the account you used to write the post with the account you wrote the answer/edit. This will let you comment and edit your own question. Best of luck.2012-08-26

2 Answers 2