A quite short proof of Bertrand's postulate

In the Wikipedia article about Bertrand's postulate, three proofs are mentioned: Chebyshev's proof, Ramanujan's proof and Erdős's proof. In this post, I give a simplified version of Chebyshev's proof.

A proof of Bertrand's postulate by Sylvester

In his paper "On Tchebycheff's Theory of the Totality of the Prime Numbers Comprised within Given Limits" from 1881, Sylvester gives an outline of a proof of Bertrand's postulate. Namely, he writes "the formulae above indicated would suffice to prove M. Bertrand's postulate, and would lead to an equation somewhat simpler in form than that led to by M. Tchebycheff's process". In this post, I provide more detail on this proof.

Numerical sum over primes

In his Mémoire Chebyshev uses bounds he found on the function \( \theta(x) =\sum_{p\le x} \log p\) to calculate numerically sums over primes \begin{equation*} \sum_{p =2 }^{\infty} f(p) \end{equation*} I explain his method in this post; to simplify the presentation, I use less sophisticated bounds on \( \theta(x) \) than Chebyshev. As an application, I calculate numerically the sum \begin{equation*} \sum_{p =2 }^{\infty} \frac{1}{p \log p} \end{equation*} with explicit bound on the obtained error.

Sign patterns of the Möbius function

I test some of the statements in the paper "Sign patterns of the Liouville and Möbius function". On page 6 in the paper one can find "the three events \( \mu(n) = +1, \mu(n) = 0, \mu(n) = -1 \) occur with asymptotic probability \( \frac{1}{2 \zeta(2)}, 1 - \frac{1}{\zeta(2)}, \frac{1}{2 \zeta(2)} \) respectively". Here, \( \mu \) is the Möbius function and \(\zeta \) is Riemann's zeta function. I interpret the concept of asymptotic probability in a naive way and just count these events in Mathematica for \( n \le 100 \).

Variations on a theme by Chebyshev

In his paper "Mémoire sur les nombres premiers", Chebyshev calculates a lower and upper bound on the function \begin{equation*} \psi(x) = \sum_{n \le x } \Lambda(n) \end{equation*} from an analysis of the expression \begin{equation}\label{eq:2} T(x) - T\left( \frac{x}{2} \right)- T\left( \frac{x}{3} \right)- T\left( \frac{x}{5} \right)+ T\left( \frac{x}{30} \right) \end{equation} with \( T(x) = \sum_{ 1 \le n \le x} \log n\ \). In this post, I quickly review Chebyshev's argument and then calculate which bounds on \(\psi(x)\) one can get from expressions that are similar to \eqref{eq:2}.

Lower and upper bound for Chebyshev's psi function

In this post, I prove the inequalities \begin{equation}\label{eq:1} \log 2 \cdot x - 3 \log x \le \psi(x) \le 2 \log 2 \cdot x + \frac{3}{\log 2} \log^2 x \end{equation} for all real \( x \ge 2 \). Hereby is \( \psi(x) = \sum_{n \le x } \Lambda(n) \) with \( \Lambda \) the von Mangoldt function. This is a second post motivated by Chebyshev's paper "Mémoire sur les nombres premiers" from 1852. In this paper, Chebyshev uses a more complex variant of the calculation below to obtain stronger inequalities. Similar inequalities can also be found on page 50 in Murty's book.

How not to prove Bertrand's postulate

This is a first post motivated by Chebyshev's paper "Mémoire sur les nombres premiers" from 1852. In this paper Chebyshev proves Bertrand's postulate that there is always a prime between \( a \) and \( 2a \) for all \( a \ge 2 \). Chebyshev bases his proof on inequalities for the function \begin{equation}\label{HNTPeq:1} T(x) - T\left( \frac{x}{2} \right)- T\left( \frac{x}{3} \right)- T\left( \frac{x}{5} \right)+ T\left( \frac{x}{30} \right) \end{equation} with \begin{equation*} T(x) = \sum_{ 1 \le n \le x} \log n \end{equation*} Chebyshev's paper is quite easy to read and can be found at this link. In this post I will follow Chebyshev's reasoning on an easier version of \eqref{HNTPeq:1}.