Let n be a positive integer. Determine the number of ordered pairs (a, b) of positive integers with lcm(a, b) = n as a function of τ .

Question

I'm not sure how to approach this problem.

I know that the arithmetic tau function denotes the number of positive divisors of n.

Answer 1

Suppose the prime factorization of $n$ is $\prod_j p_j^{v_j}$ and that of $a$ is $\prod_j p_j^{v_{j,1}}$ and that of $b$ is $\prod_j p_j^{v_{j,2}}.$ Then we must have $\max(v_{j,1}, v_{j,2}) = v_j.$ That means either they are both equal to $v_j$ (one possibility) or one is $v_j$ and the other one is less than $v_j$ including zero ($2v_j$ possibilities). Therefore $p_j$ contributes $2v_j+1$ possibilities, the same as the contribution of $p_j$ to the divisor count of $n^2.$ Hence the answer is $\tau(n^2).$ (Recall that $\tau(n) = \prod_j (v_j+1).$)