On the conditional bounds for Siegel zeros

TL;DR

This paper discusses bounds for Siegel zeros under conjectures related to Goldbach representations, clarifies previous work, and addresses gaps in recent generalizations of these bounds.

Contribution

The paper identifies and corrects a defect in recent generalizations of bounds for Siegel zeros, proposing new conjectures and discussions to refine these results.

Findings

Identified a defect in G. Bhowmik and K. Halupczok's work

Proposed a new conjecture to overcome the defect

Clarified the relationship between previous results and new discussions

Abstract

Under a weakened version of Hardy-Littlewood Conjecture on the number of representations in Goldbach problem, J. H. Fei proved bounds for the Siegel zeros. Recently G. Bhowmik and K. Halupczok generalized Fei's result under a weaker conjecture. In the first version of this paper on arXiv, we pointed out a defect in the paper of G. Bhowmik and K.Halupczok, and assumed a new conjecture and used new discussion to overcome this defect to recover their result. Afterwards, in the second version of their paper, G. Bhowmik and K. Halupczok assumed previous conjecture and used our new discussion to get a weaker result. But they did not mention our paper so that we have to make some explanation now.