On the fifth-power moment of \(\Delta_{(1)}(x)\) (Q6556224)

Inspired by the Dirichlet divisor problem and approximating moments of the function \(\Delta(x)\) denoting the error term of the sum \(\sum_{n\leq x}d(n)\), in the paper under review the author provides precise approximation for the fifth-power moment of the function \(\Delta_{(1)}(x)\) denoting the error term of the sum \(\sum_{n\leq x}d_{(1)}(n)\), where \(d_{(1)}(n)\) is the \(n\)-th coefficient of the Dirichlet series expansion of \((\zeta'(s))^2\). More precisely, the author shows that\N\[\N\int_1^T(\Delta_{(1)}(x))^5dx=T^{9/4}\sum_{j=0}^{10}c_j\log^j T+O(T^{261/118+\varepsilon}),\N\]\Nwhere the coefficients \(c_j\) are computable.