I'm trying to code the HPS (harmonic product spectrum) algorithm and the problem I'm facing is that using my guitar, the fundemental frequency (Here it's 82Hz) just isn't there, so is the 5th Harmonic at 410Hz (Actually it's the lowest local point). In my algorithm, I did the step of finding the max of all (Mul(harmonic-multiplications)) What pre/post-process should I do to get to 82Hz here.
Missing f0 is a common situation. Pitch could be defined the lowest common multiple of the spacing between spectral peaks, and they seem to be spaced by 82 Hz or 164 Hz here - so there's nothing wrong with your signal and spectrum. Harmonic sum or product should deal with this case. Could you please post plots of your harmonic sum or product? Which result do they give you?
You can also try looking at another pitch detection method like the Average Magnitude Difference Function or YIN.