On the differences between two kinds of mean value formulas of number-theoretic error terms (Q2876597)

Let \(d(n)\) denotes the number of positive divisors of \(n\), and let NEWLINE\[NEWLINE \Delta(x)=\sum_{n\leq x}d(n)-x\left(\log x+2\gamma-1\right). NEWLINE\]NEWLINE In the paper under review the authors obtain asymptotic formulas for the difference NEWLINE\[NEWLINE \sum_{n\leq x}\Delta(n)^k-\int_1^x\Delta(t)^k dt, NEWLINE\]NEWLINE for \(4\leq k\leq 10\). The authors do a similar work concerning the above mentioned difference replacing \(\Delta(x)\) by \(P(x)=\sum_{n\leq x}r(n)-\pi x\), where \(r(n)\) is the number of ways to write \(n\) as a sum of two squares. They also consider the differences for the mean value formulas of the error term of the Rankin-Selberg problem, concerning the \(n\)th Fourier coefficient of holomorphic cusp form of weight \(\kappa\) with respect to the full modular group \(\mathrm{SL}(2,\mathbb{Z})\). The paper contains some general results involving the differences for the discrete and continuous mean value formulas of arithmetic functions, and ends by an appendix about Hardy's results concerning these differences.