Writing product $\prod_{i=1}^m (p_i^{n_i}-1)$ as a sum

Question

Suppose I have an integer $N$ with prime decomposition $N=\prod_{i=1}^m p_i^{n_i}$. How can I write

$$\prod_{i=1}^m (p_i^{n_i}-1)$$

as a sum that only depends on $N$, and not it's prime decomposition?

Clearly we have

$$\prod_{i=1}^m (p_i^{n_i}-1) = N - \sum_{i=1}^{m} \frac{N}{p_i^{n_i}} + \sum_{\substack{i_1,i_2=1 \\ i_1\leq i_2}}^n \frac{N}{p_{i_1}^{n_{i_1}} p_{i_2}^{n_{i_2}}} \dots$$

So it seems like we could write this as something like

$$\prod_{i=1}^m (p_i^{n_i}-1) = \sum_{d|N}\mu(d)\frac{N}{f_{N}(d)}$$

where $f_N(d)$ is a function that looks something like

$$f_N(d) = \prod_{p_i|d_i} p_i^{n_i}$$

My questions is, is there already a function like $f_N(d)$ in the literature that will satisfy this? If not, is there some other way to write $\prod_{i=1}^m (p_i^{n_i}-1)$ as a sum that only depends on $N$?

Answer 1

Since the formula $$\prod_{i=1}^m (p_i^{n_i}-1) = n \prod_{i=1}^m \left(1-\frac1{p_i^{n_i}}\right)$$ resembles the totient function, I thought that this kind of generalization of totient function might have been studied somewhere. The book Sandor J., Crstici B. Handbook of number theory, vol.2, has a chapter on generalizations of totient function, so I tried to look there. I will copy the relevant part from this book below.

I read there that the product from your question is sometimes called unitary totient function and denoted $\varphi^*(n)$. There are also other arithmetical functions related to unitary divisors.

Also a paper by Eckford Cohen was mentioned as a reference. (Exact reference is given below.) In this paper we can find the following:

Corollary 2.4.1. $$\varphi^*(n)=\sum_{\substack{d\delta=n\\(d,\delta)=1}} \mu^*(d)\delta.$$ Where $\mu^*(n)=(-1)^{\omega(n)}$ is the unitary Möbius function and $\omega(n)$ denotes number of distinct prime factors of $n$.

This sum over unitary divisors can be rewritten as $$\sum_{\substack{d\mid n\\(d,\frac nd)=1}} \mu^*(d) \frac nd$$ which seems to be exactly the sum from your post. (Notice that unitary divisors are precisely the divisors of the form $p_i^{n_i}$.)

---

This is the part of Section 3.7.6 from Sandor-Crstici relevant to the question. (You can find much more facts about unitary versions of various arithmetical functions as well as further references in this book.)

The unitary analogue of $\varphi(n)$ was introduced by E. Cohen [83] as follows. Let $(a,b)^*$ denote the greatest divisor of a which is a unitary divisor of $b$ (a divisor $r$ of $b$ is called unitary, if $\left(r,\frac br\right)=1$.) If $(a,b)^*=1$, then $a$ is said to be semi-prime to $b$ Let $\varphi^*(n)$ be the number of positive integers $r\le n$, semi-prime to $n$. In fact, $$\varphi^*(n)=\sum_{d\mid n} d\mu^*(n/d) = \prod_{p^\alpha\mathrel{\|} n}(p^\alpha-1),$$ where $\mu^*(n)=(-1)^{\omega(n)}$ is the unitary Möbius function. For unitary divisors see also 1.9 of Chapter 1, 2.2.2 of Chapter 2, and 3.6.1 of Chapter 3. For the corresponding notions of bi-unitary divisors and convolution, see 1.9 of Chapter 1, and 2.2.3 of Chapter 2.

[83] Eckford Cohen: Arithmetical functions associated with the unitary divisors of an integer, Math. Z. 74(1960), 66-80; doi: 10.1007/BF01180473, eudml, MR 0112861, Zbl 0094.02601.