The mean square of $d_k(n)$

Question

Let $d_2(n)=d(n)$ be the divisor function, and let $$d_k(n)=\sum_{d_1\cdots d_k= n}1=\sum_{m\cdot l= n}d_{k-1}(m).$$ Can anyone point me to a reference to the size of the error term when approximating \[ \sum _{n\leq x}d_k(n)^2?\] Since the Dirichlet series is $\zeta (s)^{k^2}f(s)$ for $f$ essentially bounded and holomorphic for $\sigma >1/2$ I think a Perron formula argument would give an error something like $x^{2/(4+k^2)}$, or perhaps a bit better with a result on moments of $\zeta (s)$.

Answer 1

For larger $k$, the best error term you can hope for has the form:

$$\sum_{n\leq x}d_k(n)^2 = R_k(x) + O_\epsilon\left(x^{\frac{k^2-a}{k^2}+\epsilon} \right) $$ where $$R_k(x)=\sum_{m\leq k^2} x C_{m}(\log x)^{m},$$ for some constants $C_m$, and where $a>0$ is a constant. In particular, proving this for $a=1$ should not be too difficult, but proving it for $a>5$ is beyond current methods (see also the best existing bounds for the related Plitz Divisor Problem)

I don't know of a reference to this exact problem, but in Wilson's 1921 paper, Proofs of Some Formulæ Enunciated by Ramanujan, he showed that for the usual divisor function, $d(n)$,

$$\sum_{n\leq x}d(n)^r = L_r(x) + O_\epsilon\left(x^{\frac{2^r-1}{2^r+2}+\epsilon}\right)$$ where $L_r(x)$ takes a similar form to $R_k(x)$ above.

For smaller $k$ you can do better. Wilson in the same 1921 paper proved that for $k=2$, you have an error of the form $O_\epsilon\left(x^{\frac{1}{4}+\epsilon}\right)$.