In their paper "[Moyenne de certains fonctions arithmétiques sur les entiers friables](https://doi.org/10.1515/crll.2003.087)", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of integers that are sums of two squares, there exists a continuous function $\lambda$ on $[0,1]$ such that $\lambda(0)>0$ and
$$\sum_{n \leq x} \beta(n)= \int_0^{1/2} x^{1-t} \frac{\lambda(t)}{\sqrt{t}} dt+O\left(\frac{x}{L(x)^c}\right)\quad (x\geq 3).$$ 
Where $$L(x)=\exp\left(\frac{\log(x)^{3/5}}{\log_2(x)^{1/5}}\right)$$ and $c$ is a suitable positive constant. This result is obtained by using the Selberg–Delange method. My question is how to obtain an explicit expression for $\lambda$ and how to find a good bound for the sums on the right side, i.e. on $\sum_{n\le x}\beta(n)$.