New evaluations for a certain number-theoretic error term

Let \(R_{(1, 1)}(n)\) denote the coefficients of the Dirichlet series \(\zeta '(s) L'(s, \chi _{4})= \Sigma _{n= 1}^{\infty } R_{(1, 1)}(n) n^{- s}\) for \(Re s> 1\) and \(P_{(1)} (x)\) the error term of \(\Sigma _{n\le x} R_{(1, 1)}(n).\) A representation of the Chowla–Walum type formula for \(P_{(1)}(x)\) is derived. As a direct application, we shall give a new order estimate for \(P_{(1)}(x)\), which constitutes an improvement over the evaluation originating from Furuya et al. Furthermore, the asymptotic formula of the integral \(\int _{1}^{X} P_{(1)}^{k}(x) d x\) is established for \(k=3, 4\).