The following arithmetic function is studied by Zagier in connection with values at odd negative integers of zeta functions of real quadratic fields: $$e_r (n)=\sum_{\underset{|x|\leq n}{x^{2}\equiv n\pmod{4}}}\sigma_{r}(\frac{n-x^2}{4}),$$ where $\sigma_r(d)=\sum_{\ell|d}\ell^r$.
I am interested in the $p$-adic behavior of this function. I would like to know whether this function appears elsewhere? Thank you for any suggestions on how to study this function modulo p.
