Let \(f(n)\) be a nonnegative, strongly additive function, satisfying \[ \sum_{y\le p\le 2y}f(p)=o(1)\quad\text{as }y\to +\infty, \tag{*} \] where \(p\) runs through the prime numbers. Assume that \(f(p)\to 0\) monotonically as \(p\to +\infty\). For \(C>0\), let \(n_1