TITLE: To estimate $\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$
QUESTION [5 upvotes]: How may we estimate $$\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$$ where for every positive integer $m$ , $d(m)$ denotes the number of positive divisors of $m$  ?

REPLY [4 votes]: In what follows, we calculate the answer for the similar function $f(n)=d\left(n\right)^{2}$.  These methods can be extended to $d(n^2)^2$ without much change.  I originally misread the question, and as a result, did the calculation for $d(n)^2$.  
On prime powers, $f(p^{k})=(k+1)^{2},$ and most importantly, $f(p)=4$, and $f(p^k)$ is not too large. Let $$g(n)=\left(f*\mu*\mu*\mu*\mu\right)(n).$$  Then $g(p)=O\left(\frac{1}{p}\right),$ and so $$\sum_{n=1}^{\infty}\frac{|g(n)|}{n}=\prod_{p}\left(1+\frac{|g(p)|}{p}+O\left(\frac{1}{p^{2}}\right)\right)<\infty.$$ Now, since $g*1*1*1*1(n)=f(n),$ it follows that $$\sum_{n\leq x}f(n)=\sum_{abcde\leq x}g(a)=\sum_{a\leq x}g(a)\sum_{bcde\leq\frac{x}{a}}1.$$ Since $$\sum_{bcde\leq\frac{x}{a}}1=\sum_{n\leq\frac{x}{a}}d_{4}(n)=\frac{x}{a}\log^{3}\left(\frac{x}{a}\right)+O\left(\frac{x}{a}\log^{2}\left(\frac{x}{a}\right)\right),$$ we have that $$\sum_{n\leq x}f(n)=x\sum_{a\leq x}\frac{g(a)}{a}\log^{3}\left(\frac{x}{a}\right)+O\left(x\log^{2}(x)\sum_{a\leq x}\frac{|g(a)|}{a}\right),$$ and so  $$\sum_{n\leq x}f(n)\sim cx\log^{3}(x)$$ for some constant $c>0$.  consider the Dirichlet series $$F(s)=\prod_p\left(1+\frac{f(p)}{p^s}+\frac{f(p^2)}{p^{2s}}+\cdots\right).$$  Then $$\frac{F(s)}{\zeta(s)^4}=\prod_p\left(1+\frac{4}{p^s}+\frac{9}{p^{2s}}+\frac{16}{p^{3s}}+\cdots\right)\left(1-\frac{1}{p^s}\right)^4.$$  We consider $\zeta(s)^4$ since the coefficient of $p^s$ is $4$.  Letting $x=\frac{1}{p^s}$, we are trying to find a nice form for $$g(x)=(1-x)^4\sum_{k=0}^\infty (k+1)^2 x^k.$$  Using generating series, we have that $$g(x)=\frac{d}{dx}\left(x\frac{d}{dx}\sum_{k=0}^{\infty}x^{k+1}\right)=\frac{d}{dx}\left(x\frac{d}{dx}\frac{x}{1-x}\right)=\frac{x+1}{\left(1-x\right)^{3}}.$$ Hence $g(x)=1-x^{2},$ and so $$F(s)=\zeta(s)^{4}\prod_{p}\left(1-\frac{1}{p^{2s}}\right)=\frac{\zeta(s)^{4}}{\zeta(2s)}.$$
{"set_name": "stack_exchange", "score": 5, "question_id": 371195}