ON THE MEAN OF THE SHIFTED ERROR TERM IN THE THEORY OF THE DIRICHLET DIVISOR PROBLEM

摘要

Let 0 < alpha < 1. We show that Bernoulli polynomials appear in the difference Sigma(n <= x) Delta(j) (n + alpha) - integral(x)(1) Delta(j) (t) dt for j = 1,...,4. As a corollary of this fact, we get better approximations of integral(x)(1) Delta(j) (t) dt 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)