Question regarding a statement in A course in Arithmetic by Serre.

Question

Can anyone explain why the author used the derivative of the function to bound $\phi_n(s)$? It is on page 70 in the book.

Answer 1

He uses mean value theorem for vector (in your case $\mathbb{C}$) valued functions (version with inequality). If $f(t) = n^{-s} - t^{-s}$, then $$|\phi_n(s)| \leq \sup_{n\leq t\leq n+1}|f(n) - f(t)|\leq \sup_{n\leq t\leq n+1}\sup_{\xi\in [0,t]}|f'(\xi)|\cdot |t-n|\leq $$ $$ \sup_{n\leq \xi\leq n+1}|f'(\xi)|\cdot |n+1 - n| = \sup_{n\leq \xi\leq n+1}|f'(\xi)| = \frac{|s|}{\inf_{n\leq \xi\leq n+1}|\xi^{s+1}|} = \frac{|s|}{\inf_{n\leq \xi\leq n+1}\xi^{x+1}} \leq \frac{|s|}{n^{x+1}} $$