As an exercise for my Analytic Number Theory course, I need to prove, using Dirichlet hyperbola method, that:
$\sum_{n\leq x}\tau(n² + 1)= {3\over\pi}x\log(x) + O( x)$,
where $\tau(n)=\sum_{d|n}1$ is the divisor function.
We´ve already proved, using Dirichlet hyperbola method, that the leading behavior of the divisor summatory function is: $\sum_{n \leq x} \tau (n)= x\log x + x(2\gamma-1) + O(\sqrt x)$.
I have tried to rewrite the sum in this way: $\sum_{n \leq x}\tau(n²+1)= \sum_{k=1}^x(\sum_{n=1}^{k²+1} \tau(n) -\sum_{n=1}^{k²} \tau(n)) = \sum_{k=1}^x k²\log({k²+1\over k²})+\log(k²+1) +(2\gamma -1) +O(k)$
and evaluate its behavior but it seems it´s not the right way leading to the result we are looking for. I would appreciate suggestions about other solutions or remarks about what is wrong in my approach.
I thought about the way I was approaching the problem and I think it is not the correct method that would lead me to the solution.
At the moment my idea is to use the method underlying the proof of the leading behavior of the divisor summatory function, thus I need $\tau(n²+1)$ as the convolution $\tau(n²+1)= f*g(n)$ and then apply Dirichlet Hyperbola Method how it is well explained at page 74 of the following lecture notes: http://www.math.uiuc.edu/~hildebr/ant/main.pdf.
In order to develop my idea, as I have written before, I need to represent $\tau(n² +1)$ as a convolution but I can't see yet how to do it. Again, any suggestion would be really helpful and appreciated.
