# On a lemma by Newman relating summability and convergence

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?

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 $$n\rightarrow\infty$$.

• Sorry, perhaps I phrased my question a bit poorly. I understand perfectly well what Newman needs it for; I just wanted to know it there is some more general principle hiding behind (in terms of some general theory, like, I don't know, Dirichlet series or some kind of transformation) this since that lemma, as it is, seems very ad-hoc to me. Sep 24, 2018 at 21:40
• After reading your comment, I am not very sure that what you wanted to ask. If you want a general theory, then search for 'Tauberian Theorems'. Sep 24, 2018 at 22:08