A supplement to Chebotarevs density theorem
2025-04-30
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arXiv:2210.13412v2 [math.NT] 30 May 2023
A SUPPLEMENT TO CHEBOTAREV’S DENSITY THEOREM
GERGELY HARCOS AND KANNAN SOUNDARARAJAN
On the 50th anniversary of Chen’s theorem and the 100th anniversary of Chebotarev’s theorem
Abstract. Let L/K be a Galois extension of number fields with Galois group G. We show that
if the density of prime ideals in Kthat split totally in Ltends to 1/|G|with a power saving error
term, then the density of prime ideals in Kwhose Frobenius is a given conjugacy class C⊂G
tends to |C|/|G|with the same power saving error term. We deduce this by relating the poles of
the corresponding Dirichlet series to the zeros of ζL(s)/ζK(s).
1. Introduction
This note arose from an amusing observation by MathOverflow user Lucia [10], which informally
says that if the density of primes congruent to 1 modulo qtends to 1/φ(q) rapidly, then for all
(a, q) = 1 the density of primes congruent to amodulo qalso tends to 1/φ(q) rapidly. Here density
refers to the actual proportions. More precisely, for any σ>1/2, the asymptotic
(1) ψ(x;q, 1) = ψ(x)/φ(q) + Oε(xσ+ε)
implies that
(2) ψ(x;q, a) = ψ(x)/φ(q) + Oε(xσ+ε) for all (a, q) = 1.
The reason is simple. The relation (1) implies that the function
X
χmod q
χ6=χ0
L′
L(s, χ) = Z∞
1ψ(x, χ0)−φ(q)ψ(x;q, 1)s
xs+1 dx
extends analytically to the half-plane
Hσ:= {s∈C:ℜ(s)> σ}.
That is, the product of the Dirichlet L-functions L(s, χ) (χ6=χ0) has no zero or pole in Hσ. As
these L-functions are entire, none of them has a zero or pole in Hσ, and (2) follows easily.
Our goal is to show that this phenomenon persists in the context of Chebotarev’s density the-
orem [4,5] (cf. [11, Ch. VII, Th. 13.4]), even though the underlying Artin L-functions are only
conjectured to be entire (Artin’s conjecture).
Corollary. Let L/K be a Galois extension of number fields, with Gdenoting the Galois group.
Suppose σ>1/2is such that for any ε > 0and x>2the asymptotic formula
X
N(p)6x
Frob(p)={1}
log N(p) = 1
|G|X
N(p)6x
log N(p) + O(xσ+ε)
2020 Mathematics Subject Classification. Primary 11R42; Secondary 11M41.
Key words and phrases. Chebotarev density theorem, Artin L-functions, Heilbronn characters.
The first author was supported by the R´enyi Int´ezet Lend¨ulet Automorphic Research Group and NKFIH (National
Research, Development and Innovation Office) grant K 143876. The second author was supported in part by a grant
from the National Science Foundation, and a Simons Investigator Award from the Simons Foundation.
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arXiv:2210.13412v2[math.NT]30May2023ASUPPLEMENTTOCHEBOTAREV’SDENSITYTHEOREMGERGELYHARCOSANDKANNANSOUNDARARAJANOnthe50thanniversaryofChen’stheoremandthe100thanniversaryofChebotarev’stheoremAbstract.LetL/KbeaGaloisextensionofnumberfieldswithGaloisgroupG.WeshowthatifthedensityofprimeidealsinKthatsplitto...
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时间：2025-04-30
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