On the conditional bounds for Siegel zeros

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.