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.