PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

On generalizations of the Titchmarsh divisor problem

Volume 193 / 2020

Akshaa Vatwani, Peng-Jie Wong Acta Arithmetica 193 (2020), 321-337 MSC: Primary 11N37; Secondary 11N25. DOI: 10.4064/aa180324-23-7 Published online: 14 February 2020


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.


  • Akshaa VatwaniDepartment of Mathematics
    Indian Institute of Technology Gandhinagar
    Palaj, Gandhinagar, Gujarat 382355, India
  • Peng-Jie WongDepartment of Mathematics and Computer Science
    University of Lethbridge
    Lethbridge, Alberta T1K 3M4, Canada

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image