ISL 1989 Problem 97

An arithmetic function is a real-valued function whose domain is the set of positive integers. Define the convolution product of two arithmetic functions $f$ and $g$ to be the arithmetic function $f * g$ , where $$(f * g)(n) = \sum_{ij=n} f(i) \cdot g(j),$$ and $f^{*k} = f * f * \ldots * f$ ( $k$ times) We say that two arithmetic functions $f$ and $g$ are dependent if there exists a nontrivial polynomial of two variables $P(x, y) = \sum_{i,j} a_{ij} x^i y^j$ with real coefficients such that

$$P(f,g) = \sum_{i,j} a_{ij} f^{*i} * g^{*j} = 0,$$

and say that they are independent if they are not dependent. Let $p$ and $q$ be two distinct primes and set

$$f_1(n) = \begin{cases} 1 & \text{ if } n = p, 0 & \text{ otherwise}. \end{cases}$$

$$f_2(n) = \begin{cases} 1 & \text{ if } n = q, 0 & \text{ otherwise}. \end{cases}$$

Prove that $f_1$ and $f_2$ are independent.