-1

Given the sum

$$g(n)=\sum_{a \in \mathbb{Z}^{*}_n}a$$

where $\mathbb{Z}^{*}_n$ is the multiplicative modular unit group.

I want to know:

  • is there some standard function or symbol representing it?
  • any remarkable properties
  • any relations with arithmetic functions or Dirichlet convolutions

Particularly I have found that this number is always a multiple of $n$ and $\phi(n)$, euler's totient function.

This is exactly:

$$g(n) = \sum_{a \in \mathbb{Z}^{*}_n}a=\frac{ \phi(n)n}{2}$$

Examples:

if $n=12$: $$g(12) = 1+5+7+11 = 4*12/2 = 24$$

if $n=7$, this is the sum of the first 6 positive integers: $$g(7) = 1+2+...+6 = 6*7/2 = 21$$

Bill Dubuque
  • 282,220
David Lemon
  • 214
  • 1
    Not sure what you are asking. Are you asking for a proof that $g(n)=\frac {n\varphi(n)}2$? – lulu Jul 22 '24 at 16:52
  • @lulu Not really, it seems not difficult to prove. Im looking for some results using it or how it can be related to other functions – David Lemon Jul 22 '24 at 18:39
  • I agree, it's easy to prove. But...you already have the function described in terms of familiar arithmetic functions. What more do you want? – lulu Jul 22 '24 at 18:50
  • @lulu Anything more, it seems an interesting fact and I cant recall it in the number theory book I've read. So it seems strange havent heard about. Also I want to go in the direction to characterize a sum of integers below a given one, in terms of arithmetic functions like this, like the sum of divisors. For primes I have it. – David Lemon Jul 22 '24 at 20:26
  • 1
    A more general formula is $\sum_{a \in \mathbb{Z}^{*}n}f(a)=\sum{d|n}\mu(d)\cdot \sum_{l=1}^{n/d}f\left(l\cdot d\right)$ where $\mu$ is the Mobius function. Is it helpfull for you? – zhrd Jul 23 '24 at 05:28
  • @zhrd yes, nice formula. I also found that its not a multiplicative function – David Lemon Jul 23 '24 at 07:19

1 Answers1

1

The idea is like

  1. If $(a,n)=1$, so is $(-a,n)$. Therefore if $a\in\mathbb{Z}_n^*$, so is $n-a$, and they will always have a sum of $n$.
  2. $\mathbb{Z}_n^*$ has exactly $\phi(n)$ elements by definition.
Angae MT
  • 2,206