A Dedekind Psi Function Inequality

TL;DR

This paper proves that the Dedekind psi function attains its extreme values at primorial integers and establishes a universal inequality relating psi(N_k) to the iterated logarithm of N_k, valid for large primorials.

Contribution

It demonstrates that the Dedekind psi function's maximum values occur at primorial integers and establishes an unconditional inequality involving psi(N_k) and the double logarithm.

Findings

psi(N_k) > c * loglog N_k for large N_k, with c ≈ 1.08

The maximum of psi(n) occurs at primorial integers

The inequality holds unconditionally for all large primorials

Abstract

This note shows that the Dedekind psi function achieves its extreme values on the subset of primorial integers N_k = 2*3*5*...*p_k, where p_k is the kth prime. In particular, the inequality psi(N_k) > cloglog N_k, where c = 1.08... is a universal constant, holds for all large primorial integers N_k = 2*3*5*...*p_k unconditionally.