TITLE: Sum of $\sum_{k=1}^nd(k^2)$
QUESTION [15 upvotes]: There is a literature dealing with
$$
\sum_{k\le x}d(f(k))
$$
where $f$ is an irreducible polynomial and $d(n)$ is the number of divisors of $n$. Erdos 1952 shows that the sum $\asymp x\log x,$ which was improved to $Ax\log x+O(x\log\log x)$ by Bellman-Shapiro (cited in Scourﬁeld). But these results only apply to irreducible polynomials.

What asymptotics are known for $\sum_{k\le x}d(k^2)$?
Are there good methods for calculating this sum quickly?

The literature includes: Dirichlet 1850, Voronoi 1903 and van der Corput 1922, Kolesnik 1969, Huxley 1993, Nowak 2001 (linear); Scourﬁeld 1961, Hooley 1963, McKee 1995, McKee 1997, McKee 1999, Broughan 2002 (quadratic). The sequence is in the OEIS as A061503 but there is no real information there.

REPLY [3 votes]: It is interesting that $d(n^2)$ problem admits square root cancelation in the error term:
$$\sum_{n\leq x} d(n^2) = x P_2(\log x) +O\left(x^{1/2}\exp\left(-c\frac{\log^{3/5}x}{\log^{1/5}\log
x}\right)\right).$$
(See section 4.12 from Postnikov A. G. Introduction to analytic number theory. He reissues result from the article Stronina M. I.
Integral points on circular cones. (Russian)
Izv. Vysš. Učebn. Zaved. Matematika 1969.)