An analogue to a conjecture of S. Chowla and H. Walum (Q798359)

The author considers an analogue, in the case of the Piltz divisor problem for three dimensions, to a conjecture of Chowla and Walum which itself corresponds to the Dirichlet divisor problem, and proves the estimate \[ (*)\quad G_{a,b;k}(x):=\sum_{n\leq x^{1/3}}\sum_{n<m\leq (x/n)^{1/2}}n^ a m^ b B_ k(\{x/nm\})=O(x^{(a+b+1)/3+\epsilon}) \] for every \(\epsilon >0\), where \(\{\) \(a\}\) denotes the fractional part of \(\alpha\), and \(B_ k(x)\) denotes the k-th Bernoulli polynomial. Generalization of (*) to higher dimension would be interesting, but the author states that it appears hopeless as far as no counterpart of Vinogradov's shifting interval lemma is known.