The non-symmetric divisor function \(\tau(1,1,2;n)\) in arithmetic progression. (Q1398601)

Let \(\tau (1,1,2;n)\) denote the number of representations of \(n\) in the form \(n=n_1n_2n_3^2\) and \(D(1,1,2;X,q,a)\) the summatory function of \(\tau (1,1,2;n)\) over those \(n\) in the arithmetic progression \(a(\bmod q)\), i.e., \[ D(1,1,2;X,q,a)=\sum_{n\leq X, n\equiv a(\bmod q)}\tau (1,1,2;n). \] In this paper the author proves for any \(\varepsilon >0\), when \(X\) tends to infinity, then uniformly in \(a, q\) and \(X\), the asymptotic estimate \[ D(1,1,2;X,q,a)=\frac{c}{\phi (q/\delta)} X+ O_\varepsilon (X^{\frac{55}{69}+\varepsilon}q^{-\frac{14}{23}}) \] where \(\delta = \gcd (a,q)\) and \[ c=\text{Res}_{s=1}\left\{\left(\sum_{m\geq 1,(m,q)=1} \tau (1,1,2;n)m^{-s}\right)\frac{X^{s-1}}{s}\right\}. \]