On Euler's example of a completely multiplicative function with sum 0

TL;DR

This paper extends Euler's example of a completely multiplicative function with sum zero to Beurling generalized numbers, demonstrating how to construct such functions with specific asymptotic properties.

Contribution

It generalizes Euler's formula to Beurling numbers and shows how to construct CMO functions with prescribed growth rates.

Findings

Extended Euler's formula to Beurling numbers

Constructed CMO functions with counting functions of the form D x^a

Demonstrated applications to generalized number systems

Abstract

Euler wrote a formula expressing that l(n)/n is a completely multiplicative function with sum 0 (a CMO function) , where l(n) is the completely multiplicative function equal to -1 on the prime numbers (the Liouville function). We extend this formula to Beurling generalized numbers, using a result of Diamond. As an application we show how to construct a CMO function carried by a set of integers whose counting function is of the form D x^a (1+o(1) ) for any a between 0 and 1.