Notation used in the book "Opera de Cribro" by Freidlander and Iwaniec

Question

In the book Opera de Cribro by Freidlander and Iwaniec, the notation $\tau_r(n)$ is used for a certain function related to the number of divisors less than $n$, $\tau(n)$. The only information I know of it is that $$\sum_{n\le x}\tau_r(n) = \frac{1}{(n - 1)!}x(\log x)^{r - 1} + O(x(\log x)^{r - 2})$$ and that $$\tau^2(n) = \sum_{m^2|n}\mu(m)\tau_{4}(n/m^2)$$

Answer 1

The question seems to be what the definition of $\tau_r(n)$ as used in Opera de Cribro is. (Please, correct me if I am wrong. But, this is how the question could make sense and at the relevant place in the book this seems actually not obvious.)

Recall that $\tau(n)$ is the number of divisors of $n$, this can also be expressed as saying this is the number of pairs of two natural numbers whose product is $n$, or still differently $$\zeta(s)^2= \sum_n \frac{\tau(n)}{n^s}.$$

This usual $\tau$ is $\tau_2$. And the definition you look for is (three equivalent versions):

• $\tau_r(n)$ is the number of $r$-tuples of natural numbers whose product is $n$, or
• $\zeta(s)^r= \sum_n \frac{\tau_r(n)}{n^s}$, or
• $\tau_r(n) = \binom{v_1 + r-1}{r-1}\binom{v_2 + r-1}{r-1} \dots \binom{v_s + r-1}{r-1}$ for $n= p_1^{v_1} \dots p_1^{v_s}$.

I am not sure where (if anywhere) this is in the book you read; you can find it for example in the first chapter of Kowalski and Iwaniec "Analytic Number Theory."