The definition of Arithmetic Function

Question

Consider following functions (written on Page 120 of the book "Summing It Up" by Avner Ash and Robert Gross, 2016): $$ a(n)= \begin{cases} 0 & \text{if $n$ is not prime and} \\ 1 & \text{if $n$ is prime.} \end{cases} $$ ... It turns out that $a(n)$ is not a very easy function to work with. Numbers like to be multiplied, so our next guess is to use $$ b(n) = \begin{cases} 0 & \text{ if n is not a power of a prime and} \\ 1 & \text{if n is a power of a prime.} \end{cases} $$ This is better but still not so great. The problem is that $b(n)$ does not depend on which prime $n$ is a power of, if indeed $n$ is a prime power. Finally, we try $$ \Lambda(n)=\begin{cases}0,&\text{if $n$ is not a power of a prime and}\\ \log p,&\text{if $n = p^m$ for some prime $p$.}\end{cases} $$

QUESTION:

What is the meaning of

1. $a(n)$ is not a very easy function to work with. Numbers like to be multiplied (how this affect $a(n)$)

2. The problem is that $b(n)$ does not depend on which prime $n$ is a power of, if indeed $n$ is a prime power (what $\Lambda(n)$ does different besides the output).

Answer 1

There's a saying in mathematics: "A mathematical object is what it does." The authors of your textbook are trying to find a function that does something. What that "something" is is hard to say without reading the textbook and seeing what the function $\Lambda$ does in the proof or example that the textbook outlines. But to answer your questions directly:

1. $a(n)$ is a fine function to work with. Its values on the positive integers are $0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,\dots$; finding the value of $a(n)$ for an arbitrary $n$ is a question of open research: see Primality Test - Wikipedia for more information. However, we usually don't just define functions for the fun of it. We define functions for a purpose, be that educational or to help in a proof. The quote "numbers like to be multiplied" probably means that the proof or example in question has to do with multiplying numbers. The function $a$ is not very useful in that regard; in general we don't have $a(nm)=a(n)a(m)$ or $a(nm)=a(n)+a(m)$, or any other identities like that. In fact, unless $n$ or $m$ is equal to $1$, we have that $a(nm)=0$, so $a$ isn't very useful when it comes to multiplying numbers. $b$ doesn't have any such property either, but it does have a somewhat weaker property, that $b(n^k)=(b(n))^k$ for any natural numbers $n,k$. This makes it a bit more useful when it comes to multiplication, but it's hard to know why that function was chosen without knowing more about the application. In general, when we talk about multiplying numbers, we like to choose a multiplicative function; the page I linked shows many common examples of such functions.
2. You ask what $\Lambda$ does differently besides the output. That's a bit of a complicated question because what is the difference between any two functions with the same domain, except for the outputs? In this case, in the proof or example that your textbook is going over, it's important that if a number is a prime power, we know which prime we are raising to a power. I can't tell you why that is important unless you tell me what your textbook is doing with $\Lambda$. Usually, in number theory, logarithms are usually useful for converting products into sums, using the property that $\log(mn)=\log m+\log n$. This benefit is a little diluted when we set $\Lambda(n)=0$ whenever $n$ is not a prime power, but it is still beneficial. Maybe you're testing whether two numbers $m,n$ are powers of the same prime, which is only true when $\Lambda(mn)\neq 0$. Apparently, the value of the prime is important here, which is why we use $\Lambda$ instead of $b$.

I hope this answers your question. In general, numbers don't like to be multiplied unless we care about multiplying them. And we don't care about which prime $n$ is a power of, unless there's a reason we should. $\Lambda$ is what it does, and clearly we care about these things for some yet-unknown purpose. I'm sorry I can't give you more information, but if you want to leave a comment about this I can update the answer :)