On functions without a normal order

TL;DR

This paper uses Turán's method to demonstrate that certain arithmetic functions lack a normal order, highlighting limitations in predicting their typical behavior.

Contribution

It introduces a novel application of Turán's method to identify functions without a normal order, expanding understanding of arithmetic function behavior.

Findings

Certain arithmetic functions do not have a normal order

Turán's method effectively shows the absence of normal order

Highlights limitations in normal order predictions

Abstract

The method of Tur\'an in establishing the normal order for the number of prime divisors of a number is used to show that a certain class of arithmetic functions do not have a normal order.