Mean-value theorems and extensions of the Elliott–Daboussi theorem on Beurling’s generalized integers (I)

Abstract

It has been discovered that, in my paper [3], quoted in the title of this correction, the condition (1.9) on ψ(x) of Theorem 4 and the condition in Remark 3 are incorrect in their current formulation. The mistake is originated in the condition (1.4) on the Chebyshev function ψ(x) of Theorem 2 in [2] due to an incorrect citation from [1].To derive the correction of the conditions, we begin with Beurling's theorems. Suppose that the counting function N (x) of Beurling's generalized integers N satisfieswith a constant γ > 0, real constants A 1 , . . . , A m such that A m = A > 0, and contantsis the zeta function associated with N . The Euler product formula implies that ζ(s) has no zeros for σ > 1. Beurling [1] showed that if γ > 1 + ρ/2 then the degree α(t) of a zero 1 + it of ζ(s) on the vertical line σ = 1 is a positive integer and that the total number of suchThe online version of the original article can be found under