The result of this paper may be considered as part II of our earlier paper on Titchmarsh series with the same title. The proof of the main theorem (Theorem 2) depends upon a special case of a convexity theorem of R. M. Gabriel.

The main theme of this paper is to systematize the Hardy-Landau $\Omega$ results and the Hardy $\Omega_{\pm}$ results on the divisor problem and the circle problem. The method of ours is general enough to include the abelian group problem and the results of Richert and the later modifications by Warlimont, and in fact theorem 6 of ours is an improvement of their results. All our results are effective as in our earlier paper II with the same title. Some of our results are new.

In the original paper we noticed a problem with the assertion $L(1,\chi_1)\neq0$ in Lemma 3 of Section 1. The problem is cleared up here. The assertions of the paper all remain valid.