On the mean of the shifted error term in the theory of the Dirichlet divisor problem (Q289703)

Let, as usual, NEWLINE\[NEWLINE \Delta(x) := \sum_{n\leq x}d(n) - x(\log x + 2\gamma-1) NEWLINE\]NEWLINE denote the error term in the classical Dirichlet divisor problem, where \(d(n)\) is the number of positive divisors of \(n\) and \(\gamma = - \Gamma'(1)\) is Euler's constant. The closeness between the discrete and continuous means, namely NEWLINE\[NEWLINE \sum_{n\leq x}\Delta^k(n),\qquad \int_1^x\Delta^k(t)\,dt\leqno(1) NEWLINE\]NEWLINE was investigated by several authors (\(k\) is a fixed natural number), including \textit{J. Furuya} [J. Number Theory 115, No. 1, 1--26 (2005; Zbl 1089.11055)], \textit{X. Cao} and \textit{W. Zhai} [Monats. Math. 160, 115--142 (2010; Zbl. 1220.11119)] and by the present authors [Int. J. Number Theory 10, No. 5, 1143--1170 (2014; Zbl 1312.11078)]. It transpired that the analogy between the two quantities in (1) is not ideal, namely the difference is not too small. This is presumably due to the fact that \(\Delta(x)\) has jumps if \(x\) is an integer. Thus in the present work the authors came up with the idea to investigate the closeness of NEWLINE\[NEWLINE \sum_{n\leq x}\Delta^k(n+\alpha),\qquad \int_1^x\Delta^k(t)\,dt\leqno(2) NEWLINE\]NEWLINE where \(\alpha\) is a fixed number for which \(0 < \alpha < 1\). They obtain interesting results for the values \(1\leq k\leq4\), since for larger values one lacks precise forms for either of the expressions appearing in (2). Their results, too complicated to be stated in full here, show that the sum in (2) is a much better approximation to the integral than the sum in (1). The values of \(B_k(x)\) when \(x=\alpha\) appear in their results, where \(B_k(x)\) is the classical \(k\)-th Bernoulli polynomial (\(B_1(x) = x-1/2\), \(B_2(x) = x^2 -x + 1/6\), etc.). As a particular example, they prove that NEWLINE\[NEWLINE \begin{aligned} \sum_{n\leq x}\Delta^3(n+\alpha) &= \int_1^x \Delta^3(t)\,dt -3B_1(\alpha)C_2x^{3/2}\log x\\ & - (6\gamma-2)B_1(\alpha)C_2x^{3/2} + O(x\log^5x), \end{aligned}\leqno(3) NEWLINE\]NEWLINE where \(C_2\) is an explicit constant. Formulas like (3) show that the quantities in (2) are closest when \(\alpha\) is the zero of the corresponding Bernoulli polynomial. The paper closes with a discussion of the Dirichlet series NEWLINE\[NEWLINE \sum_{n=1}^\infty \Delta^k(n)n^{-s}, NEWLINE\]NEWLINE and the corresponding integral, namely NEWLINE\[NEWLINE \qquad \int_1^\infty t^{-s}\Delta^k(t)\,dt NEWLINE\]NEWLINE for \(k=1,2\). This gives an additional insight into the subject.