On the average orders of the error term in the Dirichlet divisor problem
This paper deals with the classical error term in the Dirichlet divisor problem, namely \[ \Delta(x) = \sum_{n\leq x}d(n) - x(\log x + 2\gamma -1), \] where \(d(n)\) is the number of divisors of \(n\) and \(\gamma\) is Euler's constant. The accent is on the sums of \(\Delta^k(n)\) when \(k=2\) or \(k=3\). A general result, embodied in Lemma 1, is that \(\sum_{n\leq x} E^k(n)\) can be well approximated by \(\int_1^x E^k(u)\,\text{ d}u\), where \(E(x) = \sum_{n\leq x}f(n) - g(x)\) and \(g(x)\) (the ``main term'') is continuously differentiable. The first result, Theorem 1, states that (\(c_1,c_2\) are suitable constants) \[ \sum_{n\leq x}\Delta^2(n) = \int_1^x \Delta^2(u)\,du + \tfrac 16 x\log^2x + c_1x\log x + c_2x + R(x), \] where \(R(x) = O(x^{3/4}\log x)\) and at the same time \(R(x) =\Omega_\pm(x^{3/4}\log x)\). This improves much on G. H. Hardy's classical result (1916) that \(R(x) = O(x^{1+\varepsilon})\). Theorem 2 brings a corresponding asymptotic formula for \(\sum_{n\leq x}\Delta^3(n) \), where again the error term is determined up to the value of the numerical constants that are involved. The author also makes plausible conjectures concerning the sums is question. The author's methods in proving both theorems are, somewhat surprisingly for such strong results, completely elementary. The key idea is that \(\Delta(x)\) can be expressed (by Dirichlet's classical hyperbola method) explicitly in terms of the function \( \psi(x) = x - [x] - 1/2, \) where \([x]\) is the integer part of \(x\). By elementary transformations the sums investigated by the author can be transformed into sums depending on \[ \int_1^x\psi(u)\psi(u/n)\,du,\; \int_1^x\psi(u)\psi(u/n_1)\psi(u/n_2)\,du, \] where \(n,n_1,n_2\) are natural numbers subject to some natural conditions. The evaluation of the above integrals is essential. Again, it is achieved by completely elementary means, in terms of the ubiquitous \(\psi\)--function.
scientific article; zbMATH DE number 5009833