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.
The Theorem--
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 .
The first part is to assume that for some $\Re(s) > 1$, $L_n(s)=\sum_{n\le x}\lambda(n)n^{-s}= 0$, then take a sequence $t_k$ such that for $p\le x$, $p^{-s-it_k}\to -p^{-s}$ which implies that $n^{-s-it_k}\to\lambda(n)n^{-s}$ and hence $\zeta_n(z+s+it_k)\to L_n(z+s)$ uniformly around $z=0$ which means that for $k$ large enough $\zeta_n(s+it_k+z)$ has a zero near $z=0$.
The last part is that $\forall x > X,\sum_{n\le x} \lambda(n)n^{-1}\ge 0$ implies $F(s)=\int_X^\infty (\sum_{n\le x}\lambda(n)n^{-1})x^{-s}dx$ has a singularity at its abscissa of convergence $\sigma$.
Proof : if $F$ is analytic then $F(\sigma-\epsilon) = \sum_{k\ge 0}\frac{\epsilon^k}{k!} \int_X^\infty (\sum_{n\le x}\lambda(n)n^{-1})x^{-\sigma} (\log x)^k dx$ if it is finite then we can invert $\sum,\int$ obtaining that $\int_X^\infty (\sum_{n\le x}\lambda(n)n^{-1})x^{-\sigma+\epsilon}dx$ is finite.
The analyticity of the LHS shows $\sigma=1/2$.
I don't know about the converse, assuming the RH what can we say about the zeros of $\zeta_n$ and the sign of $\sum_{n\le x}\lambda(n)n^{-1}$ ?