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

Abstract

Let $0\lt \alpha \lt 1$. We show that Bernoulli polynomials appear in the difference $\sum _{n \leq x}\Delta ^j(n+\alpha )-\int _1^x \Delta ^j(t)\,dt$ for $j=1, \ldots , 4$. As a corollary of this fact, we get better approximations of $\int _1^x \Delta ^j(t)\,dt$ by using zeros of Bernoulli polynomials. For $j=1,2$, we give some interpretation of this fact by means of Dirichlet series with the coefficients $\Delta ^j(n+\alpha )$.