A generalisation of a result of de la Vallée Poussin (Q427172)

A century-old result of de la Vallée Poussin is NEWLINE\[NEWLINE\sum_{\substack{ d\leq x\\ d\equiv b NEWLINE\bmod a}}\Biggl\{{x\over d}\Biggr\}= {1-\gamma\over a} x+ O(\sqrt{x}),\quad 0\leq b< a,NEWLINE\]NEWLINE where \(\gamma\) is Euler's constant. A consequence is NEWLINE\[NEWLINE\sum_{d\leq x} 1_A(d)\Biggl\{{x\over d}\Biggr\}\sim (1-\gamma) \sum_{d\leq x} 1_A(d)\quad\text{as }x\to\infty,NEWLINE\]NEWLINE where \(A\) is a residue class modulo \(a\) and \(1_A\) its characteristic function. The author generalizes this asymptotic formula to the case \(A= g(\mathbb N)\), where \(g(t)\) is a polynomial with coefficients in \(\mathbb N_0\).