Let denote the number of ordered triples of integers whose product is n. Prove that:

HINT: Count the lattice points with positive coordinates under or on the surface defined by xyz=N.

Something I know:

for some constant c.

I can see two possible ways of aproaching this.

Method 1. If I imagine a unit cube placed onto each point (x,y,z) that lies between the curve xyz=N and the three planes x=1, y=1 and z=1 then the volume (and hence number of) these cubes will exceed the volume under the curve xyz=N. By calculus (very rusty, could be wrong) I find that the volume is . Suggesting that once I consider the negative integer solutions I would get an answer of more than plus a big O term. Wrong by a factor of 2 and I don't know how to calculate the big O terms!

Method 2.

where [N/z] represents the integer part of N/z. But I have had no luch working through the detail of this idea either!