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.

Variation on a theme by Mertens

One of the formulas that Mertens proves in his paper is

\begin{equation}\label{eq1} \sum_{ p \le x } \frac{ \log p}{p} = \log x + R \quad\text{ with }\quad | R | \le 2
\end{equation}

I use Mertens method to prove the variant

\begin{equation}\label{eq2} \sum_{ n \le x } \frac{ \Lambda(n)}{n} = \log x + R \quad \text{ with } \quad -1 \le R \le 2
\end{equation}

Here, \( \Lambda \) is the von Mangoldt function. Equation \eqref{eq2} is thus similar to \eqref{eq1}, the sum is over prime powers instead of primes. It turns out that it is easier to prove \eqref{eq2} than \eqref{eq1}, because including the prime powers actually reduces the amount of estimates one has to make. The proof of \eqref{eq2} serves as a light version of the proof of \eqref{eq1} and gives insight into how the proof of \eqref{eq1} is organized.

Mertens, Ein Beitrag zur analytischen Zahlentheorie, 1874

I started reading the article in the title to see how Mertens proves his famous theorems.

Introduction

Mertens says he will prove the formulas
$$\sum_{ p \le x } \frac{1}{p}= \log\log x + C$$ and $$\prod_{ p \le x } \frac{1}{1 – \frac{1}{p}} = C’ \log x$$ The sum and the product are over primes \( p \). Mertens will also calculate the constants \( C\) and \( C’\). Mertens also says that the formulas are already in a paper by Chebyshev, but with doubtful proof. These formulas are now known as Mertens theorems.

Upper and lower bounds on the totient summatory function

In this post I use manipulations as in Ramanujan's proof of Bertrand's postulate to calculate explicit upper and lower bounds on the totient summatory function \( \Phi(x) = \sum_{n \le x} \phi(n) \). This shows how Ramanujan's proof works in a simpler situation.

Ramanujan's proof of Bertrand's postulate, 1919

I read Ramanujan's proof of Bertrand's postulate. I liked the paper because with only 2 pages it is very short. Secondly, the paper contains explicit upper and lower bounds on some arithmetical functions; such bounds can be tested in Mathematica, whereas the more common statements involving the big-O notation cannot. Murty refers to Ramanujan's proof on page 38 in his book, but Murty rephrases Ramanujan's proof with the big-O notation. I find it refreshing to read the original version of the proof instead. This post contains thoughts on the structure of Ramanujan's proof.

"On Liouville's Function" by Lehman, 1960

I calculate some of the results in the paper On Liouville's Function, R. Sherman Lehman, Math. Comp. 14 (1960), 311-320 with Mathematica. I find this paper interesting because it is an old paper that used a computer to find an explicit counterexample to a conjecture of Polya. I also find it interesting because thinking about how to compute number theoretic functions helps to understand the formulas better. The names of the sections below are the same as in the paper. Equation numbers refer to the equations in the paper.