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) \equal{} \sum_{ij\equal{}n} f(i) \cdot g(j), and f^{*k} \equal{} 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) \equal{} \sum_{i,j} a_{ij} x^i y^j with real coefficients such that
P(f,g) \equal{} \sum_{i,j} a_{ij} f^{*i} * g^{*j} \equal{} 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) \equal{} \begin{cases} 1 & \text{ if } n \equal{} p, \\
0 & \text{ otherwise}. \end{cases}
f_2(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} q, \\
0 & \text{ otherwise}. \end{cases}
Prove that f1 and f2 are independent. functionalgebradomainreal analysispolynomialalgebra unsolved