On Kronecker's lemma

While reading some parts of the nice lecture notes on Analytic Number Theory by Hildebrand, I encountered Kronecker's lemma on page 58.

If $f : \mathbb N \to \mathbb C$ is a function, and $s \in \mathbb C$ with $\Re(s) > 0$ is a complex number such that the Dirichlet series $L(f,s) = \displaystyle\sum_{n=1}^{\infty} \frac{f(n)}{n^s}$ converges, then $\displaystyle\sum_{n \le x } f(n) = o(x^s) \quad\text{for}\quad x \to + \infty$

As an example, take $f(n) = 1$, then $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{s}}$ converges for $s > 1$ and indeed $\displaystyle\sum_{n \le x } 1 = \lfloor x \rfloor = o(x^s)$ for all $s > 1$.

As a second example, take $f(n) = \frac{1}{n}$, then $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+s}}$ converges for $s > 0$ and indeed $\displaystyle\sum_{n \le x } \frac{1}{n} = \log x = o(x^s)$ for all $s > 0$.

What I like about Kronecker's lemma is that its statement and proof involve real analysis only; one does not need to know anything about the behaviour of the function $L(f,s)$ for complex numbers $s$. On the other hand, in analytic number theory, one often uses Perron's formula to estimate sums $\sum_{n \le x } f(n)$ for number theoretic functions $f$. One then needs to have information about the zeros of the Dirichlet series, and its behaviour for large imaginary values of $s$ to make Perron's formula work. All this information is thus not needed when using Kronecker's lemma.

Although a proof of Kronecker's lemma can be found in Hildebrand's lecture notes, I write here a proof for the case $s \in \mathbb R$. This proof is based on the French Wikipedia article about Kronecker's lemma and is very transparent.

Tauberian Theorem for Cesàro mean

Because the proof of the prime number theorem is related to Tauberian theorems of various kinds (see for example this paper by Mueger), I decided to prove a Tauberian theorem for a very simple case, namely the case of Cesàro means. It seems difficult to find the proof online, I therefore write one down here.