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Let $n=p^{\alpha_1}_1 \cdots p^{\alpha_m}_m,$ and define $$\lambda_k(n)= (-1)^{ [\frac{\Omega(n)}{k} ]},$$ where $\Omega(n)= \alpha_1 + \cdots + \alpha_k,$ and $[\cdot]$ is the floor function. For $...

My opinion is ; We may use id(d)=d arithmetic function and log*id dirichlet convolution in the question. i thought that ; when we multiply and divide n with $(\log d) / d$ we obtain $F(S)=\sum_{n=...