A note on the distribution of \(d_3(n)\) in arithmetic progressions

Nguyen has shown that on averaging over \(a=1,...,q\) $a=1,...,q$ the 3-fold divisor function has exponent of distribution 2/3, following Banks et al. (Int Math Res Not 1:1–25, 2005). We follow (Blomer, in: Q J Math 59:275–286, 2008) which leads to stronger bounds.