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arxiv: 1907.02914 · v2 · pith:OVVSH2Z5new · submitted 2019-07-05 · 🧮 math.NT

M\"obius formulas for densities of sets of prime ideals

Michael Kural , Vaughan McDonald , Ashwin Sah This is my paper

Pith reviewed 2026-05-25 01:53 UTC · model grok-4.3

classification 🧮 math.NT
keywords Möbius functionnatural densityprime idealsnumber fieldChebotarev densitySato-TateBeatty sequence
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The pith

If a set of prime ideals has natural density δ then δ equals the limit of a signed Möbius sum over ideals whose smallest-norm prime factor lies in the set.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that in any number field the natural density of an arbitrary set S of prime ideals can be recovered from a limit involving the generalized Möbius function. The sum is restricted to ideals whose prime factor of smallest norm belongs to S. This identity extends earlier formulas that were known only for Chebotarev sets and supplies explicit expressions for densities arising in Sato-Tate distributions and Beatty sequences. A reader cares because the result converts existence of a density into a concrete arithmetic limit that can be studied or approximated directly.

Core claim

Let K be a number field and let S be any set of prime ideals of its ring of integers that possesses a natural density δ(S). Let D(K,S) be the set of all nonzero ideals whose unique prime divisor of minimal norm lies in S. Then the limit as X tends to infinity of the negative sum, over all a in D(K,S) with norm at most X, of μ(a) divided by the norm of a equals δ(S), where μ denotes the generalized Möbius function on ideals.

What carries the argument

The auxiliary collection D(K,S) of ideals whose smallest-norm prime divisor lies in S, on which the generalized Möbius function is summed to produce the density.

If this is right

  • The formula supplies explicit Möbius expressions for densities of prime ideals lying in any Sato-Tate interval of a fixed elliptic curve.
  • Densities of primes belonging to Beatty sequences such as those near multiples of π are recovered by the same signed sum.
  • The identity holds for every number field K, not merely the rationals.
  • Any set of prime ideals that happens to possess a natural density satisfies the Möbius formula.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Direct numerical evaluation of the partial sums for moderate X could yield practical approximations to densities that are hard to compute otherwise.
  • The same interchange of limits might produce parallel formulas in settings that possess both a density notion and a compatible Möbius function, such as function fields.
  • If density is replaced by a weaker averaging procedure, an analogous identity could hold after a suitable modification of the limiting process.

Load-bearing premise

The set S must possess a natural density that exists as an ordinary limit, and the partial Möbius sums must behave regularly enough for the limit to pass inside the density definition.

What would settle it

Choose a concrete number field and a concrete set S whose density is known by other means, compute the partial Möbius sums up to successively larger X, and check whether they converge to the known density value.

read the original 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 \[ -\lim_{X \to \infty}\sum_{\substack{2 \le \operatorname{N}(\mathfrak{a})\le X\\ \mathfrak{a} \in D(K,S)}}\frac{\mu(\mathfrak{a})}{\operatorname{N}(\mathfrak{a})} = \delta(S), \] where $\mu(\mathfrak{a})$ is the generalized M\"obius function and $D(K,S)$ 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 Beatty sequences such as $\lfloor\pi n\rfloor$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper generalizes Möbius-type density formulas previously known for Chebotarev sets to arbitrary sets S of prime ideals in the ring of integers of a number field K that possess a natural density δ(S). It asserts that -lim_{X→∞} ∑_{2≤N(a)≤X, a∈D(K,S)} μ(a)/N(a) = δ(S), where D(K,S) consists of those integral ideals whose prime ideal of minimal norm lies in S, and applies the formula to densities of primes in Sato-Tate intervals for elliptic curves and in Beatty sequences.

Significance. If the stated limit formula holds, the result supplies a uniform arithmetic expression for natural densities of prime ideals via a Möbius sum, extending the Alladi–Dawsey–Sweeting–Woo line of work from special sets to general density sets. The applications to Sato-Tate and Beatty-sequence densities illustrate concrete utility in arithmetic statistics and Diophantine approximation.

minor comments (2)
  1. The abstract and introduction should explicitly recall the definition of the generalized Möbius function μ on ideals of O_K (including its values on prime powers) to make the statement self-contained for readers outside analytic number theory.
  2. In the applications section, the passage from the ideal-density formula to the stated densities for elliptic curves and Beatty sequences would benefit from a short paragraph confirming that the relevant sets of primes satisfy the natural-density hypothesis.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation for minor revision. The referee's description accurately captures the generalization of the Möbius density formula and its applications.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central result equates a Möbius sum limit over the auxiliary set D(K,S) to the assumed natural density δ(S). This equality is derived via partial summation on the prime counting function for S together with Mertens-type estimates on the product over primes, all of which are standard external tools independent of the target formula. No step defines δ(S) in terms of the sum, fits a parameter to data and renames the output a prediction, or relies on a self-citation chain for a uniqueness or ansatz claim. The generalization from prior Chebotarev results is supported by external citations whose authors do not overlap with the present paper. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the existence of natural density for S and on standard properties of the ideal Möbius function in number fields; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The set S of prime ideals possesses a natural density δ(S) that exists as a limit.
    The theorem is stated conditionally on this existence.
  • standard math The generalized Möbius function μ on ideals of O_K satisfies the usual multiplicative and inclusion-exclusion properties.
    Invoked in the definition of the sum.

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discussion (0)

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Reference graph

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