An explicit relationship between the error terms in Atkinson’s formula and in the formula for the Dirichlet divisor problem

Abstract

A simple formula is derived for the difference of the functions defining the error terms in Atkinson’s formula and in the formula for the Dirichlet divisor problem. Namely, this difference equals π/2 plus a function that is simply related to the square of the Riemann zeta function. Specifically, the large-t leading-order asymptotics of this function is expressed in terms of a double sum that differs from the large-t asymptotics of (ζ(1/2+it))2 only in the factor of 1/ln⁡(2πmn/t). Interestingly, to the leading order for large t, the derivative of the above function coincides with the square of the absolute value of the Riemann zeta function. This remarkable fact implies a very simple formula for the derivative of the function defining the error term in the Dirichlet divisor problem.