I am self studying number theory from Tom M Apostol Dirichlet Series and Modular functions in number theory and I have a doubt in theorem 8.20 of the book. I am attaching images of relevant results.
My doubt is in last paragraph of proof how by integral analog of Landau theorem function on left is analytic in half plane $\sigma$ > 1/2 .
Landau theorem -
What I could deduce using Landau theorem--> The function on right is analytic. So, the function on left converges for $ \sigma$ >1 , which is not similar to result deduced by Apostol.
Can somebody please help to make deduction given in the book .