= Solution
Write $y=N^{1/\varphi(N)}$, $L=\sum_{p\le y}|g(p)|$, and $A=\sum_{p\le y}g(p)/p$. Put $I_p(n)=\mathbf1_{p\mid n}$ and $Y_p=I_p-1/p$. The <strongly additive arithmetic function> identity is $g-A=\sum_{p\le y}g(p)Y_p$.
For any tuple $p_1,\ldots,p_j$, expand $\prod_{r=1}^j(I_{p_r}-1/p_r)$ by choosing an indicator in some positions and the constant in the others. If the selected positions contain distinct <primes> whose product is $d$, then
$$
\mathbb E_N\prod_{r\text{ selected}}I_{p_r}=\frac{\lfloor N/d\rfloor}N=\frac1d+O(N^{-1}).
$$
For the corresponding product of the independent <Bernoulli random variables> $X_p$, the <expectation> is exactly $1/d$. This includes repeated <primes>, since their <indicator functions> are idempotent, and the empty selection has zero error. Summing absolute values of all expansion coefficients bounds the difference by
$$
\frac1N\prod_{r=1}^j\left(1+\frac1{p_r}\right)\le\frac{2^j}{N}\le\frac{y^j}{N}
$$
when $y\ge2$. Expanding the $j$th power and summing the absolute coefficient weights gives
$$
\mathbb E_N(g-A)^j=\mathbb E\left(\sum_{p\le y}g(p)(X_p-1/p)\right)^j+O\left(\frac{y^jL^j}{N}\right).
$$
It remains to center at the actual <expectation>. Since $\mathbb E_N I_p=\lfloor N/p\rfloor/N$, we have $|\mathbb E_Ng-A|\le L/N$. For each $n$, both $g(n)-A$ and $g(n)-\mathbb E_Ng$ have absolute value at most $L$, because all their indicator coefficients have absolute value at most one. The identity $u^j-v^j=(u-v)\sum_{r=0}^{j-1}u^{j-1-r}v^r$ therefore bounds the change in the <central moment> by $jL^j/N$.
Finally, <independence> factors each term of the independent <central moment>. If $q_1,\ldots,q_w$ are the distinct <primes> of the tuple and $a_i$ their multiplicities, \b[the desired comparison] becomes
$$
\boxed{\mathbb E_N(g-\mathbb E_Ng)^j=\sum_{p_1,\ldots,p_j\le y}\prod_{i=1}^w g(q_i)^{a_i}\mathbb E(X_{q_i}-1/q_i)^{a_i}+O\left(\frac{j y^j}{N}L^j\right).}
$$
If $y<2$, then $g=0$ and both sides vanish, so the formula remains valid. The argument also covers complex coefficients $g(p)$; a real-valued assumption is needed for the later <normal distribution> conclusion, not for this <centered moment comparison for additive arithmetic functions>.
Back to article page
