On the characterization of additive functions with quadratic arguments (Q1203449)

The author poses the following conjecture. Let \(g_ i\in\mathbb{Z}[x]\) \((i=1,\dots,m)\) be distinct irreducible polynomials and \(f\) be a completely additive function. If \(\sum^ m_{i=1} c_ i f\bigl(g_ i(n)\bigr)=o(\log n)\), with \(c_ i\in\mathbb{R}\) \((i=1,\dots,m)\), then there exists a constant \(c\in\mathbb{R}\) such that \(f(p)=c \log p\) for the primes \(p\) which divide at least one of the \(g_ i(n)\) \((i=1,\dots,m,\;n\in\mathbb{N})\). Then she proves some special cases of the above conjecture concerning quadratic polynomials \(g_ i\).