On certain integrals involving the Dirichlet divisor problem

2017/11/27 by Ivić, Aleksandar, Zhai, Wenguang
#11M06 #11N36 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1711.09589

Abstract

We prove that ∫₁^XΔ(x)Δ₃(x) dx ≪ X^13/9log^10/3X, ∫₁^XΔ(x)Δ₄(x) dx ≪_ε X^25/16+ε, where Δ_k(x) is the error term in the asymptotic formula for the summatory function of d_k(n), generated by ζ^k(s) (Δ₂(x) ≡ Δ(x)). These bounds are sharper than the ones which follow by the Cauchy-Schwarz inequality and mean square results for Δ_k(x). We also obtain the analogues of the above bounds when \D(x) is replaced by E(x), the error term in the mean square formula for |ζ(1/2+it)|.

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