Tuesday, September 29, 2015

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.

Friday, September 25, 2015

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.

Wednesday, September 16, 2015

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.

Thursday, September 10, 2015

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 \( n \le 100 \).

Saturday, September 5, 2015

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}.