On a lemma by Newman relating summability and convergence

Question

On page 73 of his book on Analytic Number Theory, Newman presents the following lemma:

Let $a_n$ be a sequence of real numbers such that $\sum_{n=1}^\infty \frac{a_n}{n}$ exists and $a_n + \log n$ is nondecreasing. Then $a_n \longrightarrow 0$.

This lemma seems a bit ad-hoc to me. It is completely mysterious where it comes from and how it can be motivated. Is there some more general version of this, perhaps some connection to the theory of Dirichlet series, that can shed some light upon this?

Answer 1

Let me walk through the book with you on this one. The goal is proving the Prime Number Theorem. The author set this up by proving the equation (13) first.

So, we need to prove that $$ b_n = \sum_{p\leq n} \frac{\log p}p -\log n $$ converges to a limit.

After a few paragraph, the author found a constant $c$ that if we prove $$ a_n = b_n -c $$ converges to zero, then (13) follows.

Now, to prove $a_n$ converges to zero, the author proves from (12) that $$ \sum_{n=1}^{\infty} \frac{a_n}n $$ converges.

Since $a_n+\log n = \sum_{p\leq n} \frac{\log p}p-c$, the sequence $a_n+\log n$ is nondecreasing.

This is where the lemma is needed.

Lemma

If $\sum_{n=1}^{\infty} \frac{a_n}n$ converges, and $a_n+\log n$ is nondecreasing, then $$a_n \rightarrow 0$$ as $n\rightarrow\infty$.