On generalizations of the Titchmarsh divisor problem
Volume 193 / 2020
Abstract
Let $\mathcal F = \{\mathcal F_m : m \in \mathbb N\}$ be a family of Galois extensions of $\mathbb Q$, and $\mathcal D =\{ \mathcal D_m \subseteq \operatorname{Gal} (\mathcal F_m/\mathbb Q): m \in \mathbb N \}$ be a family of conjugacy classes of the corresponding Galois groups. Letting $\mathcal P_m = \mathcal P(\mathcal F_m, \mathcal D_m) $ be the corresponding Chebotarev sets of primes, we build upon a generalization of the Titchmarsh divisor problem formulated by Akbary and Ghioca (2012). We consider the sum $\sum _{p \le x} \tau _{\mathcal F, \mathcal D}^{K,C}(p)$, where $ \tau _{\mathcal F, \mathcal D}^{K,C}(p)$ not only counts all occurrences of $p$ in the family $\{\mathcal P_m\}$ of Chebotarev sets, but also imposes the condition that $p$ belongs to a certain fixed Chebotarev set $\mathcal P(K,C)$.
We obtain results for this generalization in particular cases, namely when $\mathcal {F}$ is a family of cyclotomic extensions of $\mathbb Q$ and the Chebotarev set $\mathcal P$ has level of distribution $1/2$. As a special case, we obtain a version of the Titchmarsh divisor problem in arithmetic progressions, which can be viewed as a variation of a result of Felix (2012). Finally, we generalize a result due to Fiorilli (2012) to obtain a Bombieri–Vinogradov type estimate for a modified Titchmarsh divisor problem involving a truncated divisor function.
