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.

Section 1

Mertens proves that $\theta(x) < 2 x$; he says that Chebyshev gives a sharper bound which he does not need. Along the way Mertens also proves that $\psi(x) < 2 x$. I found Mertens reasoning easy to follow because it is an easy version of the reasoning of a (later) paper by Ramanujan that I read. Ramanujan's reasoning is more involved because he proves a stronger result, namely Bertrand's postulate.

Section 2

In this section Mertens proves that $\sum_{ p \le x } \frac{\log p}{p} = \log x + R$ with $| R | < 2$. This statement is now known as Mertens first theorem. The proof takes less than a page, and only uses elementary inequalities. The proof is easy to read; in the middle I found a bit of help in this paper from Mark Villarino.

Here are some graphs illustrating this theorem.

In graph above, the blue dots are the function  $\sum_{ p \le x } \frac{\log p}{p}$ and the red line is the function $\log x$.

In the graph below I illustrate that the difference is indeed less than 2.