Higher-power moments of δ(x), E(t) and P(x)

Abstract

Let δ(x denote the error term in the asymptotic formula for the divisor sum. Using a Voronoi-type formula for δ(x, we prove that \int_{x}^{2} △(x)^3dx≈cX^{7/4} and \int_{x}^{2} △(x)^4dx≈c’X^2, where c and c′ are some positive constants. The estimate for the third-power moment shows that δ3(x) is strongly biased towards the positive side. We also obtain similar results for E(t), the error term in the mean square formula for the Riemann zeta-function on the critical line, and for P(x), the error term in the circle problem.