A NOTE ON THE LEAST PRIME THAT SPLITS COMPLETELYIN A NONABELIAN GALOIS NUMBER FIELDZHENCHAO GE, MICAH B. MILINOVICH, AND PAUL POLLACK

Abstract. We prove a nontrivial estimate for the size of the least rational
prime that splits completely in the ring of integers of certain families of nonabelian Galois number fields. Our result complements results of Linnik and
Vinogradov and of Pollack who studied this problem in the quadratic andabelian number field settings, respectively.

1. Introduction
One of the classical problems in number theory is to study the distribution ofquadratic residues and non-residues modulo a prime p. I. M. Vinogradov conjectured that the least quadratic non-residue modulo p is Oε (pε ) for any ε > 0. Recall
that the least quadratic non-residue (mod p) is always a prime. Analogously, Vinogradov conjectured that the least prime quadratic residue modulo p is Oε (pε ) as
well. Neither of these conjectures has been established.
In this paper we study a generalization of the second problem, bounding theleast prime quadratic residue modulo p. The Pólya-Vinogradov inequality andSiegel’s theorem for exceptional zeros of Dirichlet L-functions imply that the leastprime quadratic residue (mod p) is Oε (p1/2+ε ) for any ε > 0. Yu. V. Linnik and
A. I. Vinogradov [22] improved this estimate to Oε (p1/4+ε ) using Burgess’s estimates for character sums in place of the Pólya-Vinogradov inequality. Due to theuse of Siegel’s theorem, the implied constant in this estimate is ineffective. Pintz[14] later developed an elementary proof of the same bound using similar tools.
Since the least prime quadratic residue modulo an odd prime p is the smallest
prime that splits completely in a quadratic number field with a discriminant of either p or 4p (depending on the residue class of p modulo 4), this problem generalizes
to number fields. Let K be a Galois extension of Q with discriminant DK and ringof integers OK . We aim to bound the least rational prime that splits completely in
OK , denoted henceforth as qK , in terms of |DK |. As we show below, there is always
a bound of the form qK  |DK |1/2+ε with an implied constant depending on ε andthe degree [K : Q] of K over Q. We refer to this as the trivial bound for this problem. For abelian extensions K over Q, Pollack [15] recently proved the nontrivial
estimate that qK [K:Q],ε |DK |1/4+ε (see also [16]). This generalizes Linnik and
Vinogradov’s estimate for the quadratic extensions of Q and the proof uses similar
2000 Mathematics Subject Classification. 11R42, 11R44, 11F66, 11M20.
Key words and phrases. primes, split completely, number fields, Dedekind zeta-function,
subconvexity.
Research of the second author was partially supported by the NSA Young Investigator Grants
H98230-15-1-0231 and H98230-16-1-0311. Research of the third author was partly supported by
NSF award DMS-1402268.
1

2

Z. GE, M. B. MILINOVICH, AND P. POLLACK

methods. Pollack’s proof relies on essentially three ingredients: the factorizationof the Dedekind zeta-function ζK (s) into Dirichlet L-functions, Burgess’s estimatesfor character sums, and Siegel’s theorem for exceptional zeros.
In this note, for certain nonabelian Galois extensions K over Q, we prove nontrivial estimates for the size of the least prime that splits completely in OK . The
methods in [22] and [15], mentioned above, do not immediately generalize to thissetting. There is a generalization of Siegel’s theorem for exceptional zeros known
as the Brauer-Siegel theorem [2]. However, when a Galois extension K/Q is nonabelian, the Dedekind zeta-function ζK (s) does not factor into a product of Dirichlet
L-functions and so we cannot use Burgess’s estimates for character sums to estimate coefficients of ζK (s). In place of Burgess’s estimates, we invoke subconvexity
estimates for ζK (s) in the discriminant aspect. Such estimates are known whenζK (s) can be expressed in terms of product of automorphic L-functions for whichlevel-aspect subconvexity bounds have been established. We give some examples offamilies of nonabelian number fields for which this is the case in the next section.
To keep things slightly more general, we state our main theorem in terms of thefollowing subconvexity hypothesis.
Hypothesis 1. Let K be a number field of fixed degree m. There exist constants
ϑ, A ≥ 0 (depending at most on m) such that for Re(s) = 1/2, we have
(1.1)

ζK (s) m |s|A |DK |1/4−ϑ .

This hypothesis is stated by Einsiedler, Lindenstrauss, Michel, and Venkatesh in[6] where examples of families of number fields satisfying Hypothesis 1 are listed.
In particular, they note that by results in [1, 3, 4, 7, 8, 11, 21] this hypothesis holds
when K/Q is contained in a ring class field of an arbitrary quadratic extension ofan arbitrary but fixed ground field F .
Assuming this Hypothesis 1, we prove the following theorem.
Theorem 1. Let K/Q be a Galois extension of degree m, and let qK denote theleast prime that splits completely in OK . Assuming Hypothesis 1, we have
(1.2)

qK  |DK |1/2−2ϑ+ε

for any ε > 0. Here the implied constant depends at most on ε and m.
The Phragmén-Lindelöf convexity principle implies that Hypothesis 1 always
holds with ϑ = 0. Therefore Theorem 1 implies that qK ε,m |DK |1/2+ε . As
mentioned above, we refer to this as the trivial bound for the least rational primethat splits completely in OK . The generalized Lindelöf hypothesis (GLH) implies
that Hypothesis 1 holds with ϑ = 1/4. Therefore, assuming GLH, Theorem 1
implies that qK ε,m |DK |ε for any ε > 0. This is the number field analogue ofVinogradov’s conjecture for the least prime quadratic residue.
For abelian extensions K/Q, the generalized Burgess method for estimating
Dirichlet L-functions (e.g. [7, 8]) implies that ϑ = 3/8 is admissible in Hypothesis
1 and hence, via Theorem 1, that qK ε,m |DK |3/8+ε . Analogously, the stronger
subconvexity estimate of Conrey and Iwaniec [4] for quadratic Dirichlet L-functions
implies that qK ε,m |DK |1/3+ε for multiquadratic extensions K/Q. While theseestimates are nontrivial, they are weaker than the results of Vinogradov and Linnik[22] and of Pollack [15]. In the next section, we use Theorem 1 to give examples of
nonabelian Galois number fields where the results in [22] and [15] do not apply.

A NOTE ON THE LEAST PRIME THAT SPLITS COMPLETELY

3

If a Galois number field K/Q has a quadratic subfield, then the implied constantin Theorem 1 is ineffective (due to the application of the Brauer-Siegel Theorem).
This is true even for the trivial bound qK ε |DK |1/2+ε for any ε > 0. Using
classical work of Stark [18] and more recent work of Soundararajan [17], our proofof Theorem 1 can be modified to give the following effective estimate for qK for alarge class of number fields.
Theorem 2. Let K be a solvable Galois number field of degree m over Q with noquadratic subfield. ThenqK ε,m |DK |1/2 (log |DK |)εfor any ε > 0 where the implied constant is effectively computable.
It would be equally natural to ask for bounds on the least prime that does not
split completely in a number field K. This problem, which is a generalization of
the classical problem of bounding the least quadratic non-residue, is of a somewhat
different nature than the one discussed in this paper and has been investigated by
Murty [12], Vaaler and Voloch [20], Li [9], Murty and Patankar [13], and Zaman[23]. As an example of what is known, Li shows in [9] that for an arbitrary number
field K (not necessarily Galois over Q) of degree m, there is a nonsplit prime
ε |DK |1/(4(m−1))+ε for any ε > 0 when m is sufficiently large; more precisely, it
is shown that there is a nonsplit prime
ε |DK |(1+ε)/(4Am (m−1)) ,√where Am ≥ 1 − 2 (m − 1)−1/2 . As a corollary of a more general result, this more
precise bound was recently improved slightly by Zaman in [23]. The Brauer-Siegel
Theorem is not needed in this setting, so the estimates in [9] and [23] are effective.
2. Examples
In this section, we exhibit explicit examples of families of nonabelian Galois
extensions K/Q for which nontrivial estimates for the size of the least prime thatsplits completely in OK can be established using Theorem 1 and the following
subconvexity estimates.
Lemma 1. Let L(s, ρ) be an Artin L-function of degree two over Q not of icosahedral type with conductor D such that the determinant character χ is primitive
modulo D. Then, for Re(s) = 1/2, we haveL(s, ρ)  |s|A |D|1/4−1/1889+ε
for any ε > 0 where A is an absolute constant.
Proof. This is due to Blomer, Harcos, and Michel [1] improving upon the earlier
work of Duke, Friedlander, and Iwaniec [5].
Lemma 2. Let χ be a primitive Dirichlet character modulo D. Then, for Re(s) =1/2, we haveL(s, χ)  |s|A |D|η+ε ,for any ε > 0 where A is an absolute constant, η = 1/6 if χ is quadratic, andη = 3/16 otherwise.

4

Z. GE, M. B. MILINOVICH, AND P. POLLACKProof. The case for quadratic Dirichlet L-functions is due to Conrey and Iwaniec[4], while the result for general characters is due to Heath-Brown [7, 8] extending
work of Burgess [3].

Example 1. Let K/Q be a degree 2n Galois extension whose Galois group isisomorphic to the dihedral group Dn , the group of symmetries of the regular n-gon.
Then we have thatqK n,ε |DK |1/2−2/1889+ε(2.1)

for any ε > 0 where the implied constant is ineffective.
When n is odd, the dihedral group Dn has 2 degree one irreducible characters, denoted χ0 and χ1 , and (n − 1)/2 degree two irreducible characters, denotedψ1 , ψ2 , . . . , ψ n−1 . Hence the induced character for K over Q can be decomposed as2χ0 + χ1 + 2ψ1 + 2ψ2 + · · · + 2ψ n−1 .
2

From this decomposition, it follows that(n−1)/2(2.2)

ζK (s) = ζ(s)L (s, χ1 )Y

L (s, ψj )2

j=1

where ζ(s) is the Riemann zeta-function, L(s, χ1 ) is a quadratic Dirichlet L-function,and the L(s, ψj ) for 1 ≤ j ≤ (n − 1)/2 are degree two Artin L-functions over Q ofdihedral type.
When n is even, the dihedral group Dn has 4 degree one irreducible characters,
denoted χ0 , χ1 , χ2 , and χ3 , and (n − 2)/2 degree two irreducible characters, denoted ψ1 , ψ2 , . . . , ψ n−2 . In this case, the induced character for K over Q can be2
decomposed as
χ0 + χ1 + χ2 + χ3 + 2ψ1 + 2ψ2 + · · · + 2ψ n−22

and we have(n−2)/2

ζK (s) = ζ(s)L (s, χ1 ) L (s, χ2 ) L (s, χ3 )Y

2

L (s, ψj )j=1

where the L(s, χi ) for i = 1, 2, 3 are quadratic Dirichlet L-functions and the L(s, ψj )for 1 ≤ j ≤ (n − 2)/2 are degree two Artin L-functions over Q of dihedral type.
We now apply the above subconvexity estimates to both cases simultaneously.
Let Dχi denote the conductor of each Dirichlet L-function L(s, χi ) and let Dψjdenote the conductor of each Artin L-function L(s, ψj ). Then, by Lemma 1 andLemma 2, we have
1/4−1/1889+εYY
ζK (s)  |s|A  |Dχi ||Dψj |2 ij
1/4−1/1889+ε

 |s|A |DK |for Re(s) = 1/2 and ε > 0 arbitrary by the Führerdiskriminantenproduktformel(conductor–discriminant formula). Hence Hypothesis 1 holds with ϑ = 1/1889 − εand therefore the estimate in (2.1) follows from Theorem 1.

A NOTE ON THE LEAST PRIME THAT SPLITS COMPLETELY

5

Example 1 is a special case of generalized dihedral (Galois) extensions of Q. We
say a group G of order 2n is a finite generalized dihedral group if G = A o (Z/2Z)
where A is a finite abelian group and Z/2Z acts on A by τ aτ = a−1 for any a ∈ A
where τ is the generator of Z/2Z. By classical representation theory, if s denotes
the number of squares in A, then there are 2n/s one dimensional representations
χi of G and (n − n/s)/2 inequivalent two-dimensional irreducible representations
ρj of G. This accounts for all irreducible representations of G.
Now if a Galois extension K over Q with [K : Q] = 2n has Gal(K/Q) ∼= G, then2n/s−1

ζK (s) = ζ(s)Y(n−n/s)/2

L(s, χi )i=1

Y

L(s, ρj )2

j=1

where the L(s, χi ) are Dirichlet L-functions and the L(s, ρj ) are degree two Artin
L-functions over Q of dihedral type (as can be seen by considering intermediate
dihedral fields between K and Q). Again applying Lemma 1 and Lemma 2 as in
Example 1, we are led to the following conclusion.
Example 2. Let K/Q be a generalized dihedral Galois extension of Q. Then we
have thatqK ε |DK |1/2−2/1889+ε
for any ε > 0, where the implied constant is ineffective.
Quaternion octic extensions of Q are an example of a family of Galois number
fields that are not generalized dihedral Galois extensions of Q.
Example 3. Let K/Q be a (Galois) quaternion octic number field. Then for any
ε > 0 we have
(2.3)

qK ε |DK |1/2−2/1889+ε ,where the implicit constant is ineffective.
As in the previous examples, we need only verify Hypothesis 1 for the Dedekind
zeta-function ζK (s). In this case, we have
(2.4)

2

ζK (s) = ζ(s)L (s, χ1 ) L (s, χ2 ) L (s, χ3 ) L (s, ψ) ,where L(s, χi ) for i = 1, 2, 3 are quadratic Dirichlet L-functions and L(s, ψ) is adegree two Artin L-function over Q of dihedral type. For details, see [10, Section 1].
Again, applying Lemma 1 and Lemma 2 to each L-function, we obtain the desired
result from the conductor–discriminant formula.
The estimates in Examples 1 and 2 can be improved when the degree [K : Q]
is large enough using explicit bounds for the least prime ideal in the Chebotarev
density theorem. For instance, if K is a degree 2n dihedral Galois extension of Q,
then we can use the recent work of Thorner and Zaman [19, Theorem 1.1] to deducean estimate of the form
(2.5)

qK  |DK |173/n+521/ϕ(n)where ϕ(n) is the Euler ϕ-function. Their theorem is more precise and we haveused a number of inequalities to deduce this slightly weaker estimate for qK . If

6

Z. GE, M. B. MILINOVICH, AND P. POLLACK

we assume n is prime, then the effective bound in (2.5) is better than (2.1) whenn ≥ 1399. If n is composite, using explicit lower bounds for ϕ(n), we can show that
(2.5) is stronger than the bound in Example 1 when n ≥ 5814. On the other hand,
we can show that if [K : Q] ≤ 2088, then it is always better to use the bounds in
Examples 1 or 2 than the actual version [19, Theorem 1.1]. We have not tried to
optimize all of these values.
3. Lemmas
In this section, we establish some lemmas that are used to prove Theorem 1.
Lemma 3. Let D be a positive integer and D∞ = {n ∈ N : p | n implies p | D}for primes p. Then, for any ε > 0, we have
X 1
√ ε D ε .
(3.1)
n
∞
n∈D

∞

Proof. Since every n ∈ D is a product of the prime divisors of D, it follows thatω(D)−1 Y 
 
X 1
Y
1
1
1
√
√
.
=
1
−
=
1
+
≤
1
+
√
√
(3.2)
p
p−1
n
2−1
∞
n∈D

p|D

p|D

Here, as usual, ω(D) denotes the number of distinct prime divisors of D. Since
ω(D)  log D/ log log D, the result follows.

Lemma 4. Let K/Q be a Galois extension of degree m and let ζK (s) denote thecorresponding Dedekind zeta-function. For Re(s) > 1, write
∞
X
rK(n)
(3.3)
ζK (s) =
.
ns
n=1
Then
(3.4)

X

1/2+ε

rK(n)  qK

|DK |εn<qK

for any ε > 0, where the implied constant depends only on ε and m.
Proof. Since K/Q is a Galois extension, we haveepOK = (p1 · · · pg )for every rational prime p where g · e · f = m and f is the relative degree of eachof the prime ideals pi over p. A prime p splits completely if and only if g = m andf = e = 1. Hence any rational prime p that does not split completely in OK must
fall into one the following cases:(i) p is ramified with e ≥ 2 and p | DK ;(ii) e = 1 and f ≥ 2.
Recalling that qK is the least prime that splits completely in OK , we see that every
prime p less than qK falls into either case (i) or (ii). Since the primes in case (i) andcase (ii) are disjoint, every positive integer n less than qK has a unique factorizationn = uv, where u is a product of primes in case (ii) and v is a product of primes incase (i). Clearly v ∈ |DK |∞ . We claim that u is power-full whenever rK(n) 6= 0.1 To
see this, first notice that rK(·) is multiplicative, being the coefficients in a Dirichlet
1We say an integer n is power-full or square-full if p | n implies that p2 | n for any prime p.

A NOTE ON THE LEAST PRIME THAT SPLITS COMPLETELY

7

series with an Euler product. If p is a prime of type (ii), then rK(p) = 0, sincef ≥ 2. So if p k n where p is of type (ii), then rK(n) = 0. Hence, if rK(n) 6= 0, thenu is power-full.
Using the factorization n = uv described above, we have
X
X
X
rK(n) ≤
rK(v)
rK(u).
v<qK
v∈|DK |∞

n<qK

u<qK /v
u power-fullSince the number of power-full integers less than x is O(x1/2 ) and the coefficientsrK(n) ε nε for any ε > 0, it follows from Lemma 3 that
X

1/2+ε

rK(n) ε qK

X

1/2+ε

v −1/2 ε qK

|DK |ε .

v<qKv∈|DK |∞

n<qK

This completes the proof of the lemma.

Lemma 5. Let K/Q be a finite extension of degree m (not necessarily Galois).
Then, assuming Hypothesis 1, we have
∞

√
X
(3.5)
rK(n) e−n/x = x Res ζK (s) + Om x |DK |1/4−ϑ .
s=1

n=1

Proof. Using the inverse Mellin transform identity
Z c+i∞
1
(3.6)
e−y =
Γ(s) y −s ds
2πi c−i∞which is valid for c > 0 and y > 0, we deduce that
Z 2+i∞∞∞
 x s XX
1−n/x
rK(n) e
=
rK(n)
Γ(s)
ds
2πi 2−i∞nn=1
n=1
Z 2+i∞ X
∞
rK(n)
1
(3.7)
Γ(s) xs ds=
2πi 2−i∞ n=1 ns
Z 2+i∞1=
ζK (s) Γ(s) xs ds
2πi 2−i∞for any x > 0. Here the interchange of summation and integration is justified by
the absolute convergence of the series in (3.3) when Re(s) = 2. Next, we shift theline of integration in the third integral on the right-hand side left from Re(s) = 2
to Re(s) = 1/2. (To justify the contour shift, we use that ζK (s) is polynomiallybounded in the vertical strip 1/2 ≤ σ ≤ 2, while |Γ(σ + it)| decays exponentially int there.) In doing so, we pass over the simple pole of ζK (s) at s = 1 and no othersingularities of the integrand. SinceRes ζK (s) Γ(s) xs = x Res ζK (s),
s=1

s=1

we deduce that
(3.8)

∞
X
n=1

rK(n) e−n/x

1
= x Res ζK (s) +s=12πi

Z 12 +i∞
1
2 −i∞

ζK (s) Γ(s) xs ds.

8

Z. GE, M. B. MILINOVICH, AND P. POLLACK

Stirling’s formula for the gamma function and Hypothesis 1 imply that the integral
on the right-hand side isZ ∞√√m x |DK |1/4−ϑ
(1+|t|)A Γ( 12 +it) dt m x |DK |1/4−ϑ .
−∞Inserting this estimate into (3.8), the lemma follows.



4. Proof of Theorem 1
We may suppose that |DK | is sufficiently large in terms of ε, since otherwise
there are only finitely many fields K in question, and the estimate of Theorem 1
is trivial. We prove Theorem 1 by contrasting
the fact that the algebraic estimate
P
in Lemma 4 indicates that the sum n<qK rK(n) is small while the combination of
the Brauer-Siegel theorem and the analytic estimate in Lemma 5 tell us that thissum must be large if qK is sufficiently large in terms of |DK |.
Fix ε > 0. Lemma 4 states that
X
1/2+ε
(4.1)
rK(n)  qK|DK |ε .
n<qK

On the other hand, by Lemma 5, we haveX
(4.2)

n≤x log x

rK(n) ≥∞
X

rK(n) e−n/x + O (x log x)ε

n=1≥ x Res ζK (s) + Om

√

s=1


x |DK |1/4−ϑ .

Here we bounded the tail of the infinite sum as follows:
X
X
rK(n) e−n/x 
nε e−n/x  (x log x)ε .
n>x log xn>x log x(To see the last estimate, put n into intervals (x log x + jx, x log x + (j + 1)x], andnote that the contribution from the jth interval is  ((x log x)ε + j ε xε )e−j .) We
now argue that (4.1) and (4.2) are incompatible if qK ≥ |DK |1/2−2ϑ+3ε . Since theBrauer-Siegel theorem [2] states that
(4.3)

Res ζK (s)  |DK |−ε ,s=1

by choosing x with qK = x log x it follows from (4.2) thatX
qK
|DK |−ε(4.4)rK(n) log qKn≤qK

if qK ≥ |DK |1/2−2ϑ+3ε . Hence, if qK ≥ |DK |1/2−2ϑ+3ε , by combining (4.1) and(4.4) we see that
X
qK
1/2+ε
rK(n)  qK|DK |ε ,|DK |−ε 
log qKn≤qK

which is not possible if |DK | is sufficiently large. Therefore it must be the case that
qK  |DK |1/2−2ϑ+3ε . Theorem 1 now follows by replacing 3ε with ε.

A NOTE ON THE LEAST PRIME THAT SPLITS COMPLETELY

9

5. Proof of Theorem 2
In this section, we indicate what changes need to be made to our proof of Theorem 1 in order to prove Theorem 2.
Sketch of the Proof. In place of Hypothesis 1, we use the very general weak subconvexity result of Soundararajan [17] which, for Galois extensions of Q, implies
that|DK |1/4
ζK (s) m,ε |s|m/4(log |DK |)1−ε
2
for Re(s) = 1/2 and any ε > 0. Using this estimate in the proof of Lemma 5, wededuce that


∞
X
√
|DK |1/4
−n/x
.
x
(5.1)
rK(n) e= x Res ζK (s) + Om,εs=1(log |DK |)1−ε
n=1
If K/Q is a solvable Galois number field with no quadratic subfield, then Stark [18]proved an effective version of the Brauer-Siegel Theorem which states that1
(5.2)
Res ζK (s) 
s=1
log |DK |where the implied constant is effectively computable. Theorem 2 now follows byusing (5.1) and (5.2) in place of Lemma 4 and (4.3) in the proof of Theorem 1. 
Acknowledgments
This project began as a result of an SEC Faculty Travel Grant that allowed
the second author to visit the University of Georgia. We thank the Southeastern
Conference for its support. We also thank Caroline Turnage-Butterbaugh, Jesse
Thorner, and the anonymous referee for a number of useful comments.
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10

Z. GE, M. B. MILINOVICH, AND P. POLLACK

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Department of Mathematics, University of Mississippi, University, MS 38677, USA
E-mail address: zge@olemiss.edu
E-mail address: mbmilino@olemiss.edu
Department of Mathematics, University of Georgia, Athens, GA 30602, USA
E-mail address: pollack@uga.edu