Recently I am reading Stein's Complex Analysis, and he is going to prove the prime number theorem after estimating the value $1/\zeta(s)$. However, I don't understand the technical details of the proof in Proposition 1.6 in chapter 7. Here are what I have before going to my problem.

  1. If $\sigma \ge1$ and $t\in\mathbb R$, then $\log\left|\zeta^3(\sigma)\zeta^4(\sigma+it)\zeta(\sigma+2it)\right|\ge1$ (Chapter 7, Corollary 1.5)
  2. If $\sigma,t\in \mathbb R$, $|t|\ge1$, $0\le\sigma_0\le1$ and $\sigma_0\le\sigma$, then for every $\epsilon>0$, there exists a constant $c_\epsilon$ such that $|\zeta(\sigma+it)|\le c_\epsilon|t|^{1-\sigma_{0}+\epsilon}$ (Chapter 6, Proposition 2.7)

Here is my question, in Chapter 7, Proposition 1.6: The book tells that the following inequality holds for $\sigma\ge1$ and $|t|\ge1$:

$$|\zeta^4(\sigma+it)|\ge c|\zeta^{-3}(\sigma)||t|^{-\epsilon}\ge c'(\sigma-1)^3|t|^{-\epsilon}$$

The first inequality is obviously concluded from (1) and (2) above. However I can't see why the second inequality holds. Can anybody give me some hints on it?

  • 1
    $\begingroup$ Hint: $\lim_{s\rightarrow 1}\zeta(s)\sim \frac{1}{1-s}$ $\endgroup$ – tired Aug 21 '15 at 14:37
  • $\begingroup$ @tired:But this is true for $\sigma$ very close to 1, but is the inequality still true for very large $\sigma$? $\endgroup$ – Y.H. Chan Aug 21 '15 at 15:05
  • $\begingroup$ This will be the smallest value $\zeta^{-1}(s)$ will take so it's sufficent for an lower bound $\endgroup$ – tired Aug 21 '15 at 15:15

Because $$\zeta\left(\sigma\right)=O\left(\left(\sigma-1\right)^{-1}\right) $$ at $\sigma\longrightarrow1 $, hence $$\zeta^{3}\left(\sigma\right)=O\left(\left(\sigma-1\right)^{-3}\right). $$


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