Explicit upper bounds for the remainder term in the divisor problem

Abstract

We first report on computations made using the GP/PARI package that show that the error term Δ(x) in the divisor problem is = M(x, 4)+ O*(0.35 x 1/4 log x) when x ranges [1 081 080, 10 10], where M(x, 4) is a smooth approximation. The remaining part (and in fact most) of the paper is devoted to showing that |Δ(x)| ≤ 0.397 x 1/2 when x ≥ 5 560 and that |Δ(x)| ≤ 0.764 x 1/3 log x when x ≥ 9 995. Several other bounds are also proposed. We use this results to get an improved upper bound for the class number of a quadractic imaginary field and to get a better remainder term for averages of multiplicative functions that are close to the divisor function. We finally formulate a positivity conjecture concerning Δ(x). © 2011 American Mathematical Society.