Correction to: “Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem” [J. Number Theory 130 (2010) 707–715]

Abstract

We discuss the main result of [1] which is concerned with the study of generalised prime systems for which the integer counting function $N_P\left(x\right)$ is asymptotically very well-behaved, in the sense that $N_P\left(x\right)=\rho x+O\left(x^{\beta }\right)$, where ρ is a positive constant and $\beta <\frac{1}{2}$. For such systems, the associated zeta function ${\zeta }_P\left(s\right)$ is holomorphic for $\sigma =\Re s>\beta$. It was claimed that for $\beta <\sigma <\frac{1}{2}$, ${\int }_0^T\left|{\zeta }_P\right(\sigma +it){|}^{2}dt=\Omega (T^{2-2\sigma -\epsilon })$ for (i) any $\epsilon >0$, and (ii) for $\epsilon =0$ for all such σ except possibly one value.

The proof of these statements contains a flaw however, and in this Corrigendum we indicate where the mistake occurred but show that the proof can be rectified to still obtain (i) and get a slightly weaker result for (ii). The resulting Corollary 2 of [1] concerning the Dirichlet divisor problem for generalised integers remains essentially correct.