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Unusual functional equation

Source: Science ON 2021 grade X/4

March 8, 2021
functional equationnumber theorymultiplicative functionsalgebra

Problem Statement

Find all functions f:Z≥1→R>0f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0} such that for all positive integers nn the following relation holds: ∑d∣nf(d)3=(∑d∣nf(d))2,\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2, where both sums are taken over the positive divisors of nn.
(Vlad Robu)
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