New evaluations for a certain number-theoretic error term

Let \(R_{(1, 1)}(n)\) $R_{(1,1)}(n)$ denote the coefficients of the Dirichlet series \(\zeta '(s) L'(s, \chi _{4})= \Sigma _{n= 1}^{\infty } R_{(1, 1)}(n) n^{- s}\) ${\zeta }^{\prime }(s)L^{\prime }(s,{\chi }_4)={\mathrm{\Sigma }}_{n=1}^{\infty }{R}_{(1,1)}(n)n^{-s}$ for \(Re s> 1\) $Res>1$ and \(P_{(1)} (x)\) $P_{(1)}(x)$ the error term of \(\Sigma _{n\le x} R_{(1, 1)}(n).\) ${\mathrm{\Sigma }}_{n\le x}{R}_{(1,1)}(n).$ A representation of the Chowla–Walum type formula for \(P_{(1)}(x)\) $P_{(1)}(x)$ is derived. As a direct application, we shall give a new order estimate for \(P_{(1)}(x)\) $P_{(1)}(x)$ , which constitutes an improvement over the evaluation originating from Furuya et al. Furthermore, the asymptotic formula of the integral \(\int _{1}^{X} P_{(1)}^{k}(x) d x\) ${\int }_1^X{P}_{(1)}^k(x)dx$ is established for \(k=3, 4\) $k=3,4$ .