Divisor function \(\tau_3(\omega)\) in arithmetic progression

Abstract. We constructed the asymptotic formula over the ring of the Gaussian integers for summatory function of the divisor function \(d_3(\omega)\) in an arithmetic progression \(N(\omega)\equiv\ell\pmod{q}\) which is non-trivial for \(q\leq{x}^{\frac{2}{7}-\varepsilon}\).

Key words and phrases. Kloosterman sum, divisor function, asymptotic formula.