Birch's theorem: if f(n) is multiplicative and has a non-decreasing normal order then f(n)=n^{alpha}

TL;DR

Birch's theorem states that a multiplicative function with a non-decreasing normal order must be a power function, and this paper reviews its proof for educational purposes.

Contribution

The paper provides a pedagogical review of Birch's theorem and presents an open problem related to it.

Findings

Proof of Birch's theorem explained step-by-step

Clarification of conditions for multiplicative functions with normal order

Open problem posed for further research

Abstract

For pedagogical purposes (inclusion in lecture notes) we review the proof of the theorem stated in the title. At the end we state a problem.