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I have a question about the definition of divisor functions when I was reading primes in tuples by Goldston, Pintz, and Yıldırım:

Let $\omega(q)$ denote the number of prime factors of a squarefree integer $q$. For any real number $m$, we define $d_m(q) = m^{\omega(q)}$. This agrees with the usual definition of the divisor functions when $m$ is a positive integer.

Can anyone tell me why this definition agrees with usual definition of divisor functions?

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1 Answer 1

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The identity $d_m(q)=m^{\omega(q)}$ holds only when $q$ is a squarefree integer. If $q=p_1\cdots p_k$ (so that $k=\omega(q)$), then there are precisely $m^k$ ordered $m$-tuples of positive integers $(d_1,\dots,d_m)$ such that $d_1\cdots d_m=q$: exactly one coordinate $d_j$ is divisible by $p_1$, exactly one is divisible by $p_2$, and so on, and these choices can be made independently.

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Thanks. Then what is the original definition of divisor function $d_m(q)$? –  fan Jun 2 '13 at 3:14
The number of ordered $m$-tuples $(b_1,\dots,b_m)$ such that $b_1\cdots b_m = q$. For example, $d_2(q)$ is the ordinary number-of-divisors-of-$q$ function. –  Greg Martin Jun 2 '13 at 3:51
I thought $d_m(q)=\sum_{d|q}d^m$. Thanks! –  fan Jun 2 '13 at 13:44
Aha, there's our problem. The function $\sigma_m(q) = \sum_{d\mid q} d^m$ is an important function, but it would be a generalized sum-of-divisors function (usual divisors, generalized weight we're adding); instead, GPY are using a generalized number-of-divisors function (generalized divisors, usual weight $1$). –  Greg Martin Jun 2 '13 at 16:57

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