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Möbius formulas for densities of sets of prime ideals

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Abstract

We generalize results of Alladi, Dawsey, and Sweeting and Woo for Chebotarev densities to general densities of sets of primes. We show that if K is a number field and S is any set of prime ideals with natural density \(\delta (S)\) within the primes, then

$$\begin{aligned} -\lim _{X \rightarrow \infty }\sum _{\begin{array}{c} 2 \le {\text {N}}(\mathfrak {a})\le X\\ \mathfrak {a} \in D(K,S) \end{array}}\frac{\mu (\mathfrak {a})}{{\text {N}}(\mathfrak {a})} = \delta (S), \end{aligned}$$

where \(\mu (\mathfrak {a})\) is the generalized Möbius function and D(KS) is the set of integral ideals \( \mathfrak {a} \subseteq \mathcal {O}_K\) with unique prime divisor of minimal norm lying in S. Our result can be applied to give formulas for densities of various sets of prime numbers, including those lying in a Sato–Tate interval of a fixed elliptic curve, and those in a Beatty sequence such as \(\lfloor \pi n\rfloor \).

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Notes

  1. Sweeting and Woo [16] call these ideals salient.

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Acknowledgements

This research was conducted at the Emory University Mathematics Research Experience for Undergraduates, supported by the NSA (grant H98230-19-1-0013) and the NSF (Grants 1849959, 1557960). We thank Harvard University and the Asa Griggs Candler Fund for their support. The authors thank Professor Ken Ono and Jesse Thorner for guidance and suggestions. We also thank Peter Humphries for pointing out some additional references.

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Kural, M., McDonald, V. & Sah, A. Möbius formulas for densities of sets of prime ideals. Arch. Math. 115, 53–66 (2020). https://doi.org/10.1007/s00013-020-01458-z

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