Erdos-Selberg Elementary PNT

Elementary Prime number Theorem
1. Selberg symmetry identity :

\displaystyle \Lambda_2(n) := \sum_{d|n} \mu(d) \log^2 \frac{n}{d}.
\displaystyle \sum_{n \le x}\Lambda_2(n) \sim 2x\log x + O(x).

Proof:
\quad \displaystyle \frac{1}{x} \sum_{n \leq x} \Lambda_2(n) = \sum_{d \leq x} \frac{\mu(d)}{d} \frac{1}{x/d} \sum_{m \leq x/d} \log^2 m.


2. Tauberian argument

Iterative arguments- Using Brun-Titchmarsh

Relation to the analytic proof and non-vanishing of zeta(s) on \sigma =1. Where are the zeroes?
Smoothing of mobius- to get these higher Von Mangoldt functions.
Relation to Chebyshev’s estimates? Approximations to mobius..
Error terms in this elementary proof- use generalized Selberg identities for \Lambda_k

Posted in $.

Leave a comment