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.