A question about Landau’s theorem for Dirichlet series and integrals

Question

A well known theorem of Landau’s for Dirichlet series and integrals goes as follows (I copy the theorem almost exactly as it appears in Ingham’s Distribution of Prime Numbers, Theorem H in Chapter V, Section 2, although this theorem appears in many books in some way or another — Montgomery-Vaughan, Apostol, etc., etc.):

If $c_n$ (or $c(x)$), is real and of constant sign for all sufficiently large $n$ (or $x$), then the real point $s=\sigma_0$ of the line of convergence of the Dirichlet’s series (or integral)

$$\sum_{n=1}^{\infty}\frac{c_n}{n^s}\mbox{ or }\int_1^{\infty}\frac{c(x)}{x^s}dx$$

is a singularity of the function $f(s)$ represented by the series of integral.

In Montgomery-Vaughan’s book, $f(s)$ is made equal to the series or the integral, period. My question is about the proof. In all presentations, the proof is carried out by contradition. The function $f$ is assumed to be analytic at $\sigma_0$ (the abscissa of convergence), and a power series is invoked which turns out to be identical to $f$ in its circle of convergence which spills over to the left of the line of convergence, contradicting the fact that the series or integral does not converge left of this line. The problem I have with this proof is, if $f(s)$ is supposed analytic at $\sigma_0$, that already implies that $f$ must be analytic on some neighborhood of $\sigma_0$, hence left of the line of convergence anyway. Why bother with the power series? There must be something fundamental in my understanding of this theorem and complex analysis that is missing and creates this confusion. Can someone please sort this out? Thank you,

Answer 1

Your assertions in the last several lines are correct, but note that they don't say anything about whether the actual Dirichlet series (or integral) converges to the left of $\sigma_0$; they just say that $f(s)$ has an analytic continuation to the left.

In the proof, $f(s)$ is replaced by its power series at some point to the right of $\sigma_0$ like $\sigma_0+1$, which involves the derivatives of $f(s)$ at that point. Then the derivatives of $f(s)$ are themselves written in terms of the Dirichlet series (or analogously for the integral, but I'll stick with the series case hereafter), so now we have a double sum. The fact that the $c_n$ are eventually of the same sign is what allows us to switch the order of those two sums. Then, the new inner sum is recognized as the power series for $n^{-s}$, so that the entire double sum turns into the Dirichlet series again. It therefore follows that the Dirichlet series itself converges to $f(s)$ everywhere in the appropriate disk centered at $\sigma_0+1$, but that disk includes real points to the left of $\sigma_0$.