A supplement to Chebotarevs density theorem

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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

χmod q

χ6=χ0

L′

L(s, χ) = Z∞

1ψ(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

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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摘要：

arXiv:2210.13412v2[math.NT]30May2023ASUPPLEMENTTOCHEBOTAREV’SDENSITYTHEOREMGERGELYHARCOSANDKANNANSOUNDARARAJANOnthe50thanniversaryofChen’stheoremandthe100thanniversaryofChebotarev’stheoremAbstract.LetL/KbeaGaloisextensionofnumberﬁeldswithGaloisgroupG.WeshowthatifthedensityofprimeidealsinKthatsplitto...

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A supplement to Chebotarevs density theorem

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