An Elementary Proof of the Prime Number Theorem based on Möbius Function

Let $\mu(n)$ denote the M\"obius function and define $M(x)= \sum_{n\leq x}^{}\mu (n)$. This paper provides an elementary proof of the prime number theorem based on Möbius function. We prove that \begin{equation*} \displaystyle\lim_{x \to +\infty}\frac{M(x)}{x}=0 \end{equation*}, which is equivalent to the prime number theorem. We use Selberg's asymptotic formula and Tatuzawa-Iseki Proposition. The latter states that for a real- or complex-valued function F defined on (1,∞), and Mangoldt function Λ, we have F (x)log x + Xn≤x F (cid:16) x n(cid:17) Λ(n) = Xd≤x µ(d)G(cid:16) x d(cid:17). Selberg's asymptotic formula states that if we define Λ2 := µ ∗ (log)2 = Λ ∗ Λ + Λlog, then we have Λ2(n)= 2x log x + O(x). Moreover, Proposition 3 gives us µ(n)(cid:22) x n(cid:23)log x n = O(log x). It follows that F (cid:16 ) x n ( c i d : 1 7 ) = Z n+1 n  Xx/(n+1 )<k<x/n 1  . t ( c i d : 1 7 ) d t + O F ( c i d : 1 6 ) x Z x 2 H ( y ) d y , where M is a constant and 𝑥₁ ,𝑥₂ are arbitrary non-negative numbers.

Elementary Proof of the Prime Number Theorem Based on Möbius Function

This paper provides an elementary proof of the prime number theorem based on Möbius function. Let $\mu(n)$ denote the M\"obius function and define $M(x)= \sum_{n\leq x}^{}\mu (n)$. We prove that \begin{equation*} \displaystyle\lim_{x \to +\infty}\frac{M(x)}{x}=0 \end{equation*}, which is equivalent to the prime number theorem.

We use Selberg's asymptotic formula and Tatuzawa-Iseki Proposition. The latter states that for a real- or complex-valued function F defined on (1,∞), and Mangoldt function Λ, we have F (x)log x + Xn≤x F (cid:16) x n(cid:17) Λ(n) = Xd≤x µ(d)G(cid:16) x d(cid:17). Selberg's asymptotic formula states that if we define Λ2 := µ ∗ (log)2 = Λ ∗ Λ + Λlog, then we have Λ2(n)= 2x log x + O(x).

Moreover, Proposition 3 gives us µ(n)(cid:22) x n(cid:23)log x n = O(log x). It follows that F (cid:16 ) x n ( c i d : 1 7 ) = Z n+1 n  Xx/(n+1 )

By combining these two formulas with some basic algebraic manipulations, we can obtain our desired result.