A new formula for Chebotarev densities

We give a new formula for the Chebotarev densities of Frobenius elements in Galois groups. This formula is given in terms of smallest prime factors $p_{\mathrm{min}}(n)$ of integers $n\geq2$. More precisely, let $C$ be a conjugacy class of the Galois group of some finite Galois extension $K$ of $\mathbb{Q}$. Then we prove that $$-\lim_{X\rightarrow\infty}\sum_{\substack{2\leq n\leq X\\[1pt]\left[\frac{K/\mathbb{Q}}{p_{\mathrm{min}}(n)}\right]=C}}\frac{\mu(n)}{n}=\frac{\#C}{\#G}.$$ This theorem is a generalization of a result of Alladi from 1977 that asserts that largest prime divisors $p_{\mathrm{max}}(n)$ are equidistributed in arithmetic progressions modulo an integer $k$, which occurs when $K$ is a cyclotomic field $\mathbb{Q}(\zeta_k)$.


Introduction and Statement of Results
It is well-known that This follows from the fact that ζ(s) has a pole at Re(s) = 1. In particular, we observe by reordering that − lim X→∞ X n=2 µ(n) n = 1.
As this work will show, this can be interpreted as the statement that 100% of integers n ≥ 2 are divisible by a prime. To make sense of this, let p min (n) (resp. p max (n)) denote the smallest (resp. largest) prime divisor of n. Alladi shows in [1] that if gcd(ℓ, k) = 1, then which, by Dirichlet's theorem on primes in arithmetic progressions and a certain duality principle stated in Section 2, implies that largest prime divisors are equidistributed in arithmetic progressions modulo k. Here we generalize this.
In order to state our results, suppose that K is a Galois extension of Q. If p is an unramified prime and p ⊆ O K is a prime ideal lying above p, then let K/Q p denote the Artin symbol (see Section 2.2). For convenience, we let Therefore, it is well-known that K/Q p = C, where C ⊂ Gal(K/Q) is a conjugacy class.
Theorem 1. Let K be a finite Galois extension of Q with Galois group G = Gal(K/Q), and let C ⊂ G be a conjugacy class. Then we have that Remark. The convergence of the sum in Theorem 1 is conditional, and the proof of Theorem 1 in Section 3 gives an explicit convergence rate (see equation (10)).

Remark. We can view the set
. . as a "signed probability measure" which can be used to calculate Chebotarev densities via minimal prime divisors of squarefree integers.

Examples.
a) Alladi's theorem (1) is a special case of Theorem 1 where one chooses K to be a cyclotomic field, i.e. K = Q (ζ k ), where ζ k is a primitive kth root of unity.
b) Let f (x) = x 4 + x+ 1. Then f (x) has Galois group Gal(f ) = S 4 , so in particular #Gal(f ) = 24. Let K be the splitting field of f , and define the set S := {p prime : p is unramified in K and f has no roots in Z/pZ}.
For primes p ∈ S, the reduction of f modulo p is either an irreducible quartic, which corresponds to the conjugacy class in Gal(f ) = S 4 consisting of a 4-cycle (this conjugacy class has six elements), or a product of two irreducible quadratics, which corresponds to the conjugacy class in S 4 consisting of products of two transpositions (this conjugacy class has three elements). Then the probability of an irreducible quartic contributes 6 24 to the sum, and the probability of a product of irreducible quadratics contributes 3 24 to the sum, so the theorem gives For primes p ∈ S ′ , the reduction of f modulo p is a product of a linear factor and an irreducible cubic, which corresponds to the conjugacy class in S 4 consisting of a 3-cycle (this conjugacy class has eight elements). Then the theorem gives The To prove Theorem 1, we need the following theorem which is a statement about the largest prime divisors of integers.
Theorem 2. Assume the notation and hypotheses from Theorem 1. Then we have that where k is a positive constant.
In Section 2.1, we give some results which will help bound error terms in the proofs of the above theorems, and we give the duality between largest and smallest prime divisors introduced by Alladi [1]. In Section 2.2, we state and explain the Chebotarev Density Theorem [9]. In Section 3, we prove Theorems 1 and 2.

Acknowledgements
The author would like to thank Krishnaswami Alladi for sharing his inspiring results and ideas, Ken Ono for his invaluable advice and guidance, and Jesse Thorner for his helpful comments. The author would also like to thank the reviewers for taking the time to give incredibly detailed comments; among other things, the reviewers suggested a clearer proof of Lemma 3 and a better explanation of equation (4).

Error Terms and Duality.
We require several tools in order to prove Theorems 1 and 2. First, we define the function Ψ(X, Y ) := n≤X pmax(n)≤Y 1, which counts the number of integers n ≤ X with largest prime divisor p max (n) ≤ Y . Let S(X, Y ) denote the set of such integers n ≤ X with p max (n) ≤ Y . Then clearly |S(X, Y )| = Ψ(X, Y ).
We now state a theorem of Hildebrand [4] which improves an asymptotic bound for Ψ(X, Y ) given by de Bruijn [2]. We must first define the Dickman function ρ(β) as the continuous solution of the system Theorem (Hildebrand). We have that The following unpublished theorem of Maier [7] can be recognized as a corollary of Hildebrand's Theorem, and this corollary will be sufficient to prove the theorems in this paper.
The O-constant depends on ǫ, and we will later choose ǫ = 2/3 in the proof of Theorem 2. To obtain (4), we require an upper bound for ρ(β) as recommended by one of the reviewers. Norton [8,Lemma 4.7] gives the following bound, .
Applying Stirling's formula to (5), we see that From this asymptotic estimate and the above corollary, it is straightforward to see that (4) holds. These estimates will be useful in bounding error terms in the proof of Theorem 1. The following lemma will also be useful in obtaining estimates.
Lemma 3. For a ≤ X and S(X, Y ) defined as above, we have that Proof. Looking at the (Stieltjes) integral on the left hand side, a given integer n occurs whenever t is in the range t ≥ a, t ≥ p max (n), and t ≤ X/n. Therefore the integer n contributes to the integral the length of the interval from max p max (n), a to X/n, provided that max p max (n), a ≤ X/n. This is precisely the contribution of n to the sum on the right hand side.
In addition to error bounds, the proof of Theorem 1 requires the following beautiful result of Alladi on the duality between largest and smallest prime factors.
Theorem (Alladi [1]). If f is a function defined on integers with f (1) = 0, then This theorem provides the connection between Theorems 1 and 2.
2.2. The Chebotarev Density Theorem. Our main result is closely related to the Chebotarev Density Theorem, which we carefully state here. We must first give all of the machinery required to define the Artin symbol. Let L/K be a finite Galois extension of number fields, and let O L and O K be the corresponding rings of integers. Let p be any nonzero prime (maximal) ideal in O K . Then the ideal generated in O L by p can be uniquely split into distinct maximal ideals P j lying over p in the following way: there exists an integer g ≥ 1 such that We say the ideal p is unramified in L if e j = 1 for all 1 ≤ j ≤ g, which occurs for all but finitely many prime ideals. Define the absolute norm of a nonzero ideal a of the ring of integers O F of some number field F by Nm(a) := [O F : a] = |O F /a| .
For any prime ideal P lying over p, the Artin symbol L/K P is defined as the unique element σ ∈ G such that σ(α) = α Nm(p) (mod P) for all α ∈ L.
All of the prime ideals P j lying over p are isomorphic by elements of G and L/K τ (P) = τ L/K P τ −1 for τ ∈ G, so there exists a conjugacy class C associated to p such that each L/K P j lies in C. We define the Artin symbol L/K p to be the conjugacy class C.
We now define density. Let K be a number field, let Q(K) be some set of prime ideals of O K , and let P (K) be the set of all prime ideals of O K . The natural density of Q(K) is defined by provided the limit exists. For convenience, define π C (X, L/K) := #{p ∈ P C : Nm(p) ≤ X}.
In other words, π C (X, L/K) is the number of nonzero prime ideals p of O K which are unramified in L and for which Nm(p) ≤ X and L/K p = C, where C is a conjugacy class of the Galois group G = Gal(L/K).
We may now recall the Chebotarev Density Theorem [9]. Let L/K be a finite Galois extension of number fields, and let C be a conjugacy class of Gal(L/K). Let Then, as X → ∞, we have that In other words, the natural density of P C in {p ∈ P (K) : p is unramified in L} exists and is equal to #C #Gal(L/K) .
In particular, a more precise formulation of the Chebotarev Density Theorem from Lagarias and Odlyzko [5, Theorems 1.3 and 1.4] is: Theorem (Lagarias-Odlyzko [5]). For sufficiently large X ≥ c 1 (D L , n L ), where the constant c 1 depends on both the discriminant D L and the degree n L of L, we have for constants c 2 and c 3 .
Note that Li(X) := X 2 dt/ log t. Remark. The constant c 1 can be made explicit using Theorems 1.3 and 1.4 of [5].
The above theorem is useful in bounding error terms in the proofs of Theorems 1 and 2.

Proofs
The proofs of Theorems 1 and 2 closely follow the proofs of the corresponding theorems in [1]. Note that c 4 , . . . , c 12 are positive constants which will not be specified in the following proofs.
Proof of Theorem 2. We first rewrite the desired sum in terms of the function Ψ(X, Y ), for which we have asymptotic bounds.
Notice that this sum can be broken up into a sum over small primes and a sum over large primes, so that We now estimate S 1 and show that it is much smaller than S 2 . This implies that S 1 is not the main term in the asymptotic formula in equation (3), so we only need to obtain an upper bound. We see that Then we have that If Y = X 1/β for some β, then β = (log X) 1/3 . Thus, by equation (4), we have that We will now estimate S 2 , and it turns out that this will provide the main term in the asymptotic formula in equation (3). To obtain the main term, it will be convenient to define which means that Our goal now is to show that S 3 is small compared to S 2 . By the definition of Ψ X t , t , we replace it with a function counting elements of S X t , t to obtain Applying Lemma 3 and switching the order of summation in the first term, we then have that Li(X/n) + #C #G Li max p max (n), exp (log X) 2/3 , and

Combining these estimates gives
Thus we obtain the desired asymptotic formula, As a consequence of Theorem 2, we have the following lemma.
Lemma 4. Assume the notation and hypotheses from Theorem 1. Then we have that where k is a positive constant.
Proof of Lemma 4. Define the function f by Then ψ f (X) counts the number of integers n ≤ X such that K/Q pmax(n) = C, so by Theorem 2 we have that where e f (X) = O X exp −k(log X) 1/3 . The function ψ f (X) is a type of "stair-step function," meaning it oscillates as (the integral part of) X increases depending on the values of p max (n) for n ≤ X. Then we can rewrite which by Theorem 2 is where c 1 = −e f (1) 1 + By equation (9), we also have that