The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan:
The proof is very much ambiguous to me. For example:
1- The claim $\int_1^x u^{-\sigma} du = (x^{1- \sigma} -1)/(1- \sigma) < x^{1- \sigma}/(1- \sigma)$ is true for any $ \sigma$. Why the book says it is valid for $0 \le \sigma \le 1 - 1/ \log x$?
2- Why for $\sigma \ge 1 + 1/ \log x$ the integral is $< \int_1^{\infty} u^{- \sigma} du$?
3- I cannot derive (1.29), i.e. where the coefficient $(x^{1- \sigma} + 1)$ comes from?