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

Abstract

Let

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

“…

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.

…”

“…with M > M 0 should replace the one appearing in the condition (1.4) of Theorem 2 of [2] and hence the condition (1.9) of Theorem 4 and the one in Remark 3 of [3]. With the new condition (2), Lemma 3.2 of [2] can be upheld by the argument given in the correction of [2] and no more change in [2,3] is needed.…”

“…With the new condition (2), Lemma 3.2 of [2] can be upheld by the argument given in the correction of [2] and no more change in [2,3] is needed.…”