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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

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.

Authors

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

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