EXPLICIT MERTENS’ THEOREMS FOR NUMBER FIELDS AND
DEDEKIND ZETA RESIDUE BOUNDS WITH GRH
STEPHAN RAMON GARCIA AND ETHAN SIMPSON LEE
Abstract. Assuming the Generalized Riemann Hypothesis we obtain uni-
form, effective number-field analogues of Mertens’ theorems, each with explicit
error bounds given in terms of the degree and discriminant of the number field.
To this end, we establish explicit bounds for the residue of the Dedekind zeta-
function at s = 1 by obtaining an explicit version of Duke’s short-sum theorem
for entire Artin L-functions.
1. Introduction
Assuming the Generalized Riemann Hypothesis, our aim is to establish uniform,
explicit number-field analogues of Mertens’ classical theorems:∑p≤x
log p
p= log x+O(1), (1)
∑p≤x
1
p= log log x+M +O
(1
log x
), (2)
∏p≤x
(1− 1
p
)=
e−γ
log x
(1 + o(1)
). (3)
Here p denotes a prime number, M = 0.2614 . . . is the Meissel–Mertens constant,
and γ = 0.5772 . . . is the Euler–Mascheroni constant. Mertens obtained these
results in 1874 [21], well before the prime number theorem was proved indepen-
dently by Hadamard [7] and de la Vallee Poussin [2]. See Ingham [9, Thm. 7] or
Montgomery–Vaughan [22, Thm. 2.7] for modern proofs of Mertens’ theorems, and
Rosen [28, (3.17) - (3.30)] for explicit unconditional error bounds.
Rosen [27, Lem. 2.3, Lem. 2.4, Thm. 2] generalized (1), (2) and (3) to the number-
field setting without explicit error terms; see also [15]. Assuming GRH, we provide
generalizations of (1), (2) and (3) in the number-field setting with explicit error
bounds. Our results are uniform (valid for x ≥ 2) and the constants involved depend
only on the degree and discriminant of the number field. This complements [4], in
which the authors obtained similar results in the unconditional context.
2010 Mathematics Subject Classification. 11N32, 11N05, 11N13.Key words and phrases. Number field, discriminant, Generalized Riemann Hypothesis, GRH,
Mertens’ theorem, prime ideal, prime counting function, Chebotarev density theorem, Artin L-
function, character.
SRG supported by NSF Grant DMS-1800123.
1
2 STEPHAN RAMON GARCIA AND ETHAN SIMPSON LEE
Definitions. Let K be a number field with ring of algebraic integers OK. Let nKdenote the degree of K and ∆K the absolute value of the discriminant of K. In what
follows, p ⊂ OK denotes a prime ideal, a ⊂ OK an ideal, and N(a) the norm of a.
The Dedekind zeta-function
ζK(s) =∑a⊆OK
1
N(a)s=∏p
(1− 1
N(p)s
)−1corresponding to K is analytic on C with exception of a simple pole at s = 1. The
analytic class number formula asserts that the residue of ζK(s) at s = 1 is
κK =2r1(2π)r2hKRK
wK√
∆K, (4)
where r1 is the number of real places of K, r2 is the number of complex places of K,
wK is the number of roots of unity in K, hK is the class number of K, and RK is the
regulator of K [14]. The nontrivial zeros of ζK lie in the critical strip, 0 < Re s < 1,
where there might exist an exceptional zero β, which is real and not too close to
Re s = 1 [31, p. 148]. The Generalized Riemann Hypothesis (GRH) claims that the
nontrivial zeros of ζK(s) satisfy Re s = 12 ; in particular, β does not exist.
Let K ⊆ L be a Galois extension of number fields with Galois group G =
Gal(L/K). Let nL denote the degree of L over K and let ∆L denote the abso-
lute value of the discriminant of L over Q. Suppose that P is a prime ideal of Llying above an unramified prime p of K. The Artin symbol [L/p] denotes the con-
jugacy class of Frobenius automorphisms corresponding to prime ideals P|p. For
each conjugacy class C ⊂ G, the prime ideal counting function is
πC(x,L/K) = #{p : p is unramified in L,
[L/Kp
]= C, NK(p) ≤ x
},
in which NK(·) denotes the norm in K. If K = L, then we may specialize our
notation so that the prime ideal counting function is πK(x) =∑N(p)≤x 1.
Statement of results. Assuming GRH, we obtain the following number-field ana-
logues of Mertens’ theorems (1), (2) and (3), each valid for x ≥ 2, with explicit
constants that depend only upon the degree and discriminant of the number field.
The proof involves a recent estimate of Grenie–Molteni [6] for πK(x).
Theorem A. Assume GRH. For a number field K and x ≥ 2,∑N(p)≤x
logN(p)
N(p)= log x+AK(x), (A1)
∑N(p)≤x
1
N(p)= log log x+MK +BK(x), (B1)
∏N(p)≤x
(1− 1
N(p)
)=
e−γ
κK log x
(1 + CK(x)
), (C1)
in which
MK = γ + log κK +∑p
[1
N(p)+ log
(1− 1
N(p)
)],
EXPLICIT MERTENS’ THEOREMS FOR NUMBER FIELDS 3
|AK(x)| ≤ 4.73 log ∆K + 9.27nK + log 2, (A2)
|BK(x)| ≤ 13.47 log ∆K + (26.37 + 0.12 log x)nK√x
, and (B2)
|CK(x)| ≤ |EK(x)|e|EK(x)| with |EK(x)| ≤ nKx− 1
+ |BK(x)|. (C2)
In particular, EK(x) = o(1) (and hence CK(x) = o(1) as x→∞),
γ + log κK − nK ≤ MK ≤ γ + log κK, (M)
and
1
e134.5nK−26.76 log log ∆K≤ κK ≤ (log log ∆K)nK−1e152.74nK−63.23 (K)
For nK = 1 (K = Q), Schoenfeld [29, Cor. 2-3] provides smaller constants. In
particular, κQ = 1 so we can largely restrict our attention to nK ≥ 2. Although
unconditional bounds for κK are available (see [17–20, 31] and the discussion in
[4]), stronger bounds are possible under GRH. To establish (K), we first prove the
following explicit version of Duke’s short-sum theorem [3, Prop. 5].
Theorem B (Explicit Duke’s Theorem). Let L(s, χ) be an entire Artin L-function
that satisfies GRH, in which χ has degree d and conductor N . Then∣∣∣∣ logL(1, χ)−∑
p≤(logN)12
χ(p)
p
∣∣∣∣ ≤ 89.5 + 134.5d. (5)
The constants in Theorems A and B are rounded here for aesthetic purposes; the
proofs exhibit these constants with greater accuracy. Without explicit constants,
one can improve upon (K) somewhat. Assuming GRH, Cho–Kim obtained(1
2+ o(1)
)ζ(nK)
eγ log log ∆K≤ κK ≤ (2 + o(1))
nK−1 (eγ log log ∆K)nK−1 (6)
for nK ≥ 2 [1]. We have chosen here to pursue an explicit version of Duke’s short-
sum theorem since that result could be of independent interest. For example,
Theorem B provides an explicit estimate for the class number hK of a totally real
number field K whose normal closure has the symmetric group SnK as its Galois
group. Indeed, the class number formula implies
hK =
√∆K
2nK−1RKL(1, χ),
in which L(s, χ) = ζK(s)/ζ(s) [3, p. 103].
Slight changes to the proof of Theorem A (see Remark 1) yield the following.
Theorem C. Assume GRH and let K ⊆ L be a Galois extension of number fields.
For x ≥ 2, ∑N(p)≤x
logN(p)
N(p)=
#C#G
log x+AL(x),
∑N(p)≤x
1
N(p)=
#C#G
log log x+ML +BL(x),
4 STEPHAN RAMON GARCIA AND ETHAN SIMPSON LEE
∏N(p)≤x
(1− 1
N(p)
)=
e−γ
κL(log x)#C/#G(1 + CL(x)
),
in which p is a prime ideal in K that does not ramify in L, κL denotes the residue of
ζL(s) at s = 1, and AL(x), BL(x), CL(x), ML are defined as AK(x), BK(x), CK(x),
MK are in Theorem A, but with nL, ∆L, κL in place of nK, ∆K, κK. Our explicit
bounds for MK and κK also carry forward for ML and κL.
Outline of the paper. Section 2 contains the proof of Theorems A and Theorem
C, except for (K). The proof of Theorem B is in Section 3, which finally yields (K).
Acknowledgements. We thank Karim Belabas, Eduardo Friedman, Lenny Fuk-
shansky, Peter Kim, Stephane Louboutin, and Tim Trudgian for helpful feedback.
2. Proof of Theorem A
We prove (A1), (B1), and (C1), along with the associated error bounds (A2),
(B2), and (C2), respectively, in separate subsections. We then prove (M), which is
a consequence of the computations in the proof of (C1). We defer the proof of (K)
until Section 3 since it uses the explicit version of Duke’s theorem (Theorem B).
2.1. Preliminaries. We require an explicit version of the prime ideal theorem,
which was originally established by Landau in 1903 (without explicit constants) [13].
Recall that πK(x) =∑N(p)≤x 1 is the prime-ideal counting function and let
Li(x) =
∫ x
2
dt
log t
denote the offset logarithmic integral. Under GRH,
πK(x) = Li(x) +RK(x), (7)
in whichRK(x) = O(x1/2 log x). To boundRK(x) explicitly, one can follow Lagarias–
Odlyzko [12]. Since the quality of the error term one obtains depends upon the
zero-free region of ζK one uses, GRH naturally enters the conversation. In 2019,
Grenie–Molteni [6, Cor. 1] proved that GRH implies that (7) holds for x ≥ 2 with
|RK(x)| ≤√x
[(1
2π+
3
log x
)log ∆K +
(log x
8π+
1
4π+
6
log x
)nK
]. (8)
This strengthens a result announced by Oesterle [24] and improves the constants
Winckler obtains [32, Thm. 1.2] by using methods from Schoenfeld [29], Lagarias–
Odlyzko [12], and previous work of Grenie–Molteni [5].
The final important ingredient we need is the following formula for a continuously-
differentiable function f . Integration by parts and (7) provide∑N(p)≤x
f(N(p)) = f(x)πK(x)−∫ x
2
f ′(t)πK(t) dt
= f(x) Li(x)−∫ x
2
f ′(t) Li(t) dt+ f(x)RK(x)−∫ x
2
f ′(t)RK(t) dt
=
∫ x
2
f(t)
log tdt+ f(x)RK(x)−
∫ x
2
f ′(t)RK(t)dt. (9)
EXPLICIT MERTENS’ THEOREMS FOR NUMBER FIELDS 5
Remark 1. We briefly remark how to adjust the proof below to obtain Theorem
C. For x ≥ 2, Grenie–Molteni [6, Cor. 1] proved
πC(x) =#C#G
Li(x) +RL/K(x),
in which
|RL/K(x)| ≤ #C#G√x
[(1
2π+
3
log x
)log ∆L +
(log x
8π+
1
4π+
6
log x
)nL
].
Since #C/#G ≤ 1, we can bound |RL/K(x)| above by (8). The proof below goes
through mutatis mutandis. For example, any occurrence of Li(x) should be replaced
by (#C/#G) Li(x).
2.2. Proof of (A1) and (A2). Let f(t) = log t/t in (9) and obtain∑N(p)≤x
logN(p)
N(p)= log x+ f(x)RK(x)−
∫ x
2
f ′(t)RK(t)dt− log 2︸ ︷︷ ︸AK(x)
. (10)
For x ≥ 2, log x/√x achieves its maximum value 2/e at x = e2 and (log x)2/
√x
achieves its maximum value 16/e2 at x = e4. Thus, (8) yields
|f(x)RK(x)| ≤ log x√x
[(1
2π+
3
log x
)log ∆K +
(log x
8π+
1
4π+
6
log x
)nK
].
=1√x
(3 log ∆K + 6nK) +log x√x
(log ∆K
2π+nK4π
)+
(log x)2√x
nK8π
≤ 1√2
(3 log ∆K + 6nK) +2
e
(log ∆K
2π+nK4π
)+
2nKe2π
< 2.23843 log ∆K + 4.3874nK. (11)
Observe that
f ′(t) =1− log t
t2
undergoes a sign change from positive to negative at x = e and that
log x
8π+
1
4π+
6
log x, whose derivative is
1
8πx− 6
x(log x)2,
decreases until e4√3π ≈ 215,328.6 (in particular, it is decreasing on [2, e]). Conse-
quently, we split the integral in (10) at x = e. First note that∣∣∣∣∫ e
2
f ′(t)RK(t) dt
∣∣∣∣≤∫ e
2
1− log t
t3/2
[(1
2π+
3
log t
)log ∆K +
(log t
8π+
1
4π+
6
log t
)nK
]dt
≤(
4√e−√
2(1 + log 2)
)[(1
2π+
3
log 2
)log ∆K +
(log 2
8π+
1
4π+
6
log 2
)nK
]< 0.14203 log ∆K + 0.27737nK. (12)
For x ≥ e, symbolic integration provides∣∣∣∣∫ x
e
f ′(t)RK(t) dt
∣∣∣∣
6 STEPHAN RAMON GARCIA AND ETHAN SIMPSON LEE
≤∫ ∞e
log t− 1
t3/2
[(1
2π+
3
log t
)log ∆K +
(log t
8π+
1
4π+
6
log t
)nK
]dt
=(3π√eEi(− 1
2 ) + 6π + 2) log ∆K
π√e
+(12π√eEi(− 1
2 ) + 24π + 7)nK
2π√e
< 2.34600 log ∆K + 4.59546nK, (13)
in which Ei(x) is the exponential integral.
In light of (10), we add (11), (12), and (13), and obtain the bound
|AK(x)| < 4.72646 log ∆K + 9.26023nK + log 2.
This concludes the proof of (A2). �
2.3. Proof of (B1) and (B2). Let f(t) = 1/t in (9) and obtain∑N(p)≤x
1
N(p)=
∫ x
2
dt
t log t+RK(x)
x+
∫ x
2
RK(t) dt
t2
= log log x− log log 2 +RK(x)
x+
∫ ∞2
RK(t) dt
t2−∫ ∞x
RK(t) dt
t2
= log log x+
∫ ∞2
RK(t)
t2dt− log log 2︸ ︷︷ ︸MK
+RK(x)
x−∫ ∞x
RK(t)
t2dt︸ ︷︷ ︸
BK(x)
,
in which the convergence of the improper integral is guaranteed by (8).
For x ≥ 2, (8) provides
|RK(x)|x
≤ 1√x
[(1
2π+
3
log x
)log ∆K +
(log x
8π+
1
4π+
6
log x
)nK
]and∫ ∞
x
|RK(t)| dtt2
≤∫ ∞x
1
t3/2
[(1
2π+
3
log t
)log ∆K +
(log t
8π+
1
4π+
6
log t
)nK
]dt
= log ∆K
∫ ∞x
1
t3/2
(1
2π+
3
log t
)dt+ nK
∫ ∞x
1
t3/2
(log t
8π+
1
4π+
6
log t
)dt
≤ log ∆K
∫ ∞x
1
t3/2
(1
2π+
3
log x
)dt+ nK
∫ ∞x
1
t3/2
(log t
8π+
1
4π+
6
log x
)dt
=1√x
[(1
π+
6
log x
)log ∆K +
(log x
4π+
1
π+
12
log x
)nK
].
For x ≥ 2, the triangle inequality provides
|BK(x)| ≤ 1√x
[(3
2π+
9
log 2
)log ∆K +
(3 log x
8π+
5
4π+
18
log 2
)nK
](14)
<1√x
[13.47 log ∆K + (26.37 + 0.120 log x)nK] .
Now we find the constant MK, following Ingham [9]. Since this part of the proof
is identical to that in [4], we present an abbreviated version here.
EXPLICIT MERTENS’ THEOREMS FOR NUMBER FIELDS 7
For Re s > 1, partial summation and (B1) provide∑p
1
N(p)s= limx→∞
( ∑N(p)≤x
1
N(p)s
)= limx→∞
( ∑N(p)≤x
1
N(p)s−1N(p)
)
= limx→∞
(1
xs−1
∑N(p)≤x
1
N(p)
)+ (s− 1)
∫ ∞2
( ∑N(p)≤t
1
N(p)
)dt
ts
= (s− 1)
∫ ∞2
( ∑N(p)≤t
1
N(p)
)dt
ts
= (s− 1)
∫ ∞2
MK
tsdt+ (s− 1)
∫ ∞2
BK(t)
tsdt+ (s− 1)
∫ ∞2
log log t
tsdt. (15)
As s → 1+, the first integral tends to MK, the second tends to zero by (B2), and
the substitution ts−1 = ey converts the third integral into
(s− 1)
∫ ∞2
log log t
tsdt =
∫ ∞log(2s−1)
e−y log y dy − 21−s log(s− 1),
which tends to −γ − log(s− 1). As s→ 1+, the Euler product formula yields
MK − γ + o(1)
= log(s− 1) +∑p
[1
N(p)s+ log
(1− 1
N(p)s
)]−∑p
log
(1− 1
N(p)s
)= log
((s− 1)ζK(s)
)+∑p
[1
N(p)s+ log
(1− 1
N(p)s
)]= log κK +
∑p
[1
N(p)+ log
(1− 1
N(p)
)]+ o(1), (16)
in which the last sum converges, by comparison with∑
pN(p)−2. �
2.4. Proofs of (C1) and (C2). The proof is identical to that in [4] so we just
sketch it here for the sake of completeness. From (16), we may write
− γ − log κK +MK =∑
N(p)≤x
[1
N(p)+ log
(1− 1
N(p)
)]+ FK(x). (17)
Since
0 ≤ −y − log(1− y) ≤ y2
1− y, (18)
it follows that
|FK(x)| = −∑
N(p)>x
[1
N(p)+ log
(1− 1
N(p)
)]≤
∑N(p)>x
1
N(p)(N(p)− 1)
≤∑p>x
∑fi
1
pfi(pfi − 1)< nK
∑m>x
1
m(m− 1)≤ nKx− 1
, (19)
8 STEPHAN RAMON GARCIA AND ETHAN SIMPSON LEE
in which∑fi
denotes the sum over the inertia degrees fi of the prime ideals lying
over p. Then (B1) and (17) imply
−γ − log κK +MK =∑
N(p)≤x
log
(1− 1
N(p)
)+ log log x+MK +BK(x) + FK(x)︸ ︷︷ ︸
EK(x)
.
Exponentiate and simplify to get∏N(p)≤x
(1− 1
N(p)
)=
e−γ
κK log xe−EK(x) =
e−γ
κK log x
(1 + CK(x)
),
where |CK(x)| ≤ |EK(x)|e|EK(x)| since |et − 1| ≤ |t|e|t| for t ∈ R. �
2.5. Proof of (M). Since there are no prime ideals of norm less than 2, (16) yields
MK = γ + log κK + FK(2− δ) for δ ∈ (0, 1),
in which FK(x) is defined in (17) In particular, (19) reveals that
|FK(2− δ)| ≤ nK as δ → 0+.
In light of (18), each summand in FK(2 − δ) is non-positive, so FK(2 − δ) ≤ 0 as
δ → 0+. Consequently, −nK ≤ FK(2− δ) ≤ 0 and hence
−nK ≤MK − γ − log κK ≤ 0,
which is equivalent to (M). �
To truly complete the proof of Theorem A, we must prove (K). This requires
Theorem B, which is our next task.
3. Proof of Theorem B
3.1. Prerequisites. Most of the relevant background for what follows can be found
in [10, Ch. 5]. Let K be a number field of degree nK ≥ 2 over Q and let L/Q denote
the Galois closure of the extension K/Q. Then we may write
L(s, χ) =ζK(s)
ζ(s)=
∞∑k=1
χ(k)
ks, for Re s > 1, (20)
in which χ is the character of a certain (n − 1)-dimensional representation of
Gal(L/Q). This representation is a direct sum of irreducible representations, none
of which are trivial. Under these circumstances, the conductor N of χ satisfies
N = ∆K [30, p. 159], [23, Cor. 11.8]. In particular, N ≥ 3 by Minkowski’s bound.
It is known that L(s, χ) is meromorphic on C; Artin’s conjecture asserts that it
is entire. The Euler product formula
ζK(s) =∏p
(1−N(p)−s
)−1, for Re s > 1,
ensures that any zeros of ζK(s) in the right half plane occur in the critical strip
0 ≤ Re s ≤ 1. GRH implies that these zeros lie on the critical line Re s = 12
(we adhere to the wide interpretation of GRH, in which one extends the Riemann
Hypothesis to all global L-functions and not merely the Dirichlet L-functions; some
call this the “Extended Riemann Hypothesis”). Unconditionally, L(s, χ) has no
EXPLICIT MERTENS’ THEOREMS FOR NUMBER FIELDS 9
zeros or poles on the vertical line Re s = 1 [10, Cor. 5.47]. Since ζ(s) has a simple
pole at s = 1 with residue 1, (20) implies that
κK = Ress=1 ζK(s) = lims→1
(s− 1)ζ(s)L(s, χ) = lims→1
L(s, χ) = L(1, χ). (21)
In particular, this ensures that L(1, χ) > 0 since ζK(s)/ζ(s) > 0 for s > 1 and
because ζK(s) has a simple pole at s = 1.
Dirichlet convolution shows that the coefficients in (20) satisfy
χ(k) =∑i|k
akµ
(k
i
),
in which ak denotes the number of ideals in K of norm k and µ is the Mobius
function. Therefore,
χ(p) = apµ(1) + a1µ(p) = ap − 1 ∈ [−1, nK − 1]. (22)
Indeed, if p is inert in K then N(p) = pnK 6= p and hence χ(p) = −1. The other
extreme λ(p) = nK − 1 occurs if p splits totally in K.
Let d denote the degree of χ and N its conductor. The function L(s, χ) satisfies
the functional equation
Λ(1− s, χ) = εχNs− 1
2 Λ(s, χ), (23)
in which
Λ(s, χ) = πds2 Γ(s
2
) d+`2
Γ(s+ 1
2
) d−`2
L(s, χ), (24)
|εχ| = 1, and ` is the value of χ on complex conjugation. Moreover, L(s, χ) is of
finite order with a canonical product of genus 1 [3, p. 112]; this is needed below for
a Phragmen–Lindelof argument.
3.2. Preparatory lemmas. Let L(s, χ) be an entire Artin L-function, in which χ
has degree d and conductor N . We assume that L(s, χ) satisfies the GRH, so that
logL(s, χ) is analytic on Re s > 1/2. First, we derive an upper bound for L(s, χ).
Lemma 2. For Re s ≥ δ > 1,
|L(s, χ)| ≤ ζ(δ)d. (25)
Proof. For Re s > 1, (20) implies
logL(s, χ) =∑p
∞∑m=1
1
mχ(pm)p−ms, (26)
in which |χ(pm)| ≤ d [3, (23), p. 111]. Therefore, (26) converges absolutely since
| logL(s, χ)| ≤∑p
∞∑m=1
1
m|χ(pm)|p−mRe s ≤ d
∑p
∞∑m=1
p−mδ
m= d log ζ(δ). (27)
It follows that for δ > 1,
|L(s, χ)| = |elogL(s,χ)| ≤ e| logL(s,χ)| ≤ ed log ζ(δ) = ζ(δ)d. �
Next, we derive an upper bound for L(s, χ) on a slightly larger half plane. As is
customary, we write s = σ + it for a complex variable, in which σ, t ∈ R.
10 STEPHAN RAMON GARCIA AND ETHAN SIMPSON LEE
Lemma 3. Let c1 = 12πζ( 3
2 ) < 4.1035. Then
|L(s, χ)| ≤ cd1N(t2 + 25
4
)d/2for σ ≥ 1
2 .
Proof. We require two gamma-function estimates from Rademacher [26, Lem. 1 &
2]. The first estimate states that for a ≥ 0 and − 12 ≤ σ ≤
12 ,∣∣∣∣Γ(a2 + 1−s
2 )
Γ(a2 + s2 )
∣∣∣∣ ≤ ( 12 |a+ 1 + s|) 1
2−σ. (28)
The second estimate states that for a− 12 ≥ 0 and − 1
2 ≤ σ ≤12 ,∣∣∣∣Γ(a2 + 1−s
2 )
Γ(a2 + s2 )
∣∣∣∣ ≤ ( 12 |a+ s|) 1
2−σ. (29)
From (23) and (24),
Λ(1− s, χ) = εχNs− 1
2πds2 Γ(s
2
) d+`2
Γ(s+ 1
2
) d−`2
L(s, χ).
Substituting 1− s and χ in (24) yields
Λ(1− s, χ) = πd(1−s)
2 Γ(1− s
2
) d+`2
Γ(
1− s
2
) d−`2
L(1− s, χ).
Equate the previous two formulas, simplify, and obtain
L(s, χ) =εχπ
d( 12−s)
Ns− 12
(Γ( 1−s
2 )
Γ( s2 )
) d+`2(
Γ(1− s2 )
Γ( s+12 )
) d−`2
L(1− s, χ).
From (28) with a = 0 and (29) with a = 1, for − 12 ≤ σ ≤
12 we have
|L(s, χ)| ≤∣∣(Nπd) 1
2−s∣∣ ∣∣∣∣Γ( 1−s
2 )
Γ( s2 )
∣∣∣∣d+`2∣∣∣∣Γ( 1
2 + 1−s2 )
Γ( 12 + s
2 )
∣∣∣∣d−`2
|L(1− s, χ)|
≤ (Nπd)12−σ
[(12 |1 + s|
) 12−σ
] d+`2[(
12 |1 + s|
) 12−σ
] d−`2
|L(1− s, χ)|
= (Nπd)12−σ
[(12 |1 + s|
) 12−σ
]d|L(1− s, χ)|.
Let σ = − 12 , use (26) to pass from χ to χ, and (25) to bound |L( 3
2 + it, χ)|. Then
|L(− 12 + it, χ)| ≤ Nπd
2d|1 + s|d|L( 3
2 + it, χ)|
≤ N(π2 ζ( 3
2 ))d|1 + s|d = cd1N |1 + s|d.
Applying the Phragmen–Lindelof theorem to the vertical strip − 12 ≤ Re s ≤ 3
2
yields |L(s, χ)| ≤ Ncd1|1 + s|d in the strip.
For − 12 ≤ σ ≤
32 , we also obtain
|1 + s|2 = |(1 + σ) + it|2 = (1 + σ)2 + t2 ≤ t2 + 254 .
It follows that
|L(s, χ)| ≤ cd1N(t2 + 25
4
) d2 for 1
2 ≤ Re s ≤ 32 .
This bound also holds for Re s > 32 by (25) since
|L(s, χ)| ≤ ζ(σ)d ≤ ζ( 32 )d ≤ cd1 ≤ cd1N
(t2 + 25
4
)d/2. �
EXPLICIT MERTENS’ THEOREMS FOR NUMBER FIELDS 11
Our next lemma gives estimates L′(s, χ)/L(s, χ) in a certain vertical strip.
Lemma 4. Let c2 = 38423π < 5.3144. Then∣∣∣∣L′(s, χ)
L(s, χ)
∣∣∣∣ ≤ c2d log(N
1d (|t|+ 4)
)for 5
8 ≤ Re s ≤ 278 .
Proof. GRH implies that logL(s, χ) is analytic on Re s > 12 . Lemma 3 gives
Re logL(s, χ) = log |L(s, χ)| ≤ log(cd1N
(254 + t2
)d/2)= d log c1 + logN + d
2 log(254 + (Im s)2
)for Re s ≥ 1
2 . If f is analytic on a neighbourhood of |z| ≤ R, [11, (3.7.12)] implies
|f ′(z)| ≤ 4R
π(R2 − |z|2)sup|ξ|=R
|Re f(ξ)− α|
for any α ∈ R. Fix t ∈ R and apply the previous inequality to
f(z) = logL(z + 2 + it, χ) = logL(s, χ)
with
R = 32 , | s− (2 + it)︸ ︷︷ ︸
z
| ≤ 118 , and α = d log c1.
For |s− (2 + it)| ≤ 118 , we obtain∣∣∣∣L′(s, χ)
L(s, χ)
∣∣∣∣ ≤ 4( 32 )
π(( 32 )2 − ( 11
8 )2) sup|ξ|= 3
2
(Re logL(s+ ξ, χ)− α
)≤ 384
23πsup|ξ|= 3
2
(logN + d
2 log(254 + Im(s+ ξ)2
) )≤ c2
(logN + d
2 log(254 + (|t|+ Im ξ)2
)≤ c2
(logN + d
2 log(254 + (|t|+ 3
2 )2)
≤ c2(
logN + d2 log((|t|+ 4)2)
)= c2d
(logN
1d + log(|t|+ 4)
)= c2d log(N
1d (|t|+ 4)).
Since t ∈ R is arbitrary, we obtain the desired bound for 58 ≤ Re s ≤ 27
8 . �
Duke’s approach to the previous lemma uses a result of Littlewood [16, Lem. 4],
which is [3, Lem. 2] in Duke’s paper, in place of the sharper [11, (3.7.12)] used
above. More information about “sharp real part theorems” for the derivative can
be found in [11, Ch. 5], along with a host of historical references.
Our next result is useful for an important step in the proof of Lemma 6.
Lemma 5. For N ≥ 1 and − 78 ≤ σ ≤ −
38 ,∫ ∞
−∞|Γ(σ + it)| log
(N
1d (|t|+ 4)
)dt ≤ c3 logN
d+ c4,
in which
c3 =2
538 e
43
7√π
< 30.1794 and c4 =2
538 e
43
7√π
(log 4− e2π Ei(−2π)
)< 46.0461.
12 STEPHAN RAMON GARCIA AND ETHAN SIMPSON LEE
Here Ei(x) denotes the exponential integral function.
Proof. For s = σ + it with σ ≥ 0, we have the inequality [25, 5.6E9]:
|Γ(s)| ≤√
2π|s|σ− 12 e−π|t|/2 exp
(1
6|s|
).
Since Γ(s+ 1) = sΓ(s) for s ∈ C, the previous inequality yields
|Γ(s)| = |Γ(s+ 1)||s|
≤√
2π|s+ 1|σ+ 12 e−π|t|/2
|s|exp
(1
6|s+ 1|
). (30)
For − 78 ≤ σ ≤ −
38 ,
√2π|s+ 1|σ+ 1
2
|s|exp
(1
6|s+ 1|
)︸ ︷︷ ︸
f(s)
≤√
2πe( 58 + |t|) 1
8
|t|→ 0
uniformly in σ as |t| → ∞, so we may use numerical methods to maximize f(s) in
the strip − 78 ≤ σ ≤ −
38 . This reveals that f(s) attains its maximum value
α =
√π 2
378 e
43
7< 23.7029
at s = − 78 . Therefore, (30) infers that
|Γ(σ + it)| ≤ αe−π|t|/2, for − 78 ≤ σ ≤ −
38 .
Because, |Γ(z)| = |Γ(z)| for z ∈ C,∫ ∞−∞|Γ(σ + it)| log
(N
1d (|t|+ 4)
)dt = 2
∫ ∞0
|Γ(σ + it)| log(N
1d (|t|+ 4)
)dt
≤ 2α
∫ ∞0
e−π|t|/2 log(N
1d (|t|+ 4)
)dt
=4α logN
πd+
4α(
log 4− e2π Ei(−2π))
π
= c3logN
d+ c4. �
3.3. Proof of Theorem B. The first step in our proof of Theorem B is Lemma
6, which is an explicit version of [3, Lem. 4].
Lemma 6. Let L(s, χ) be an entire Artin L-function that satisfies GRH, in which
χ has degree d and conductor N . For 1 ≤ u ≤ 32 and x > 1,∣∣∣∣∑
p
(log p)χ(p)p−ue−p/x +L′(u, χ)
L(u, χ)
∣∣∣∣ ≤ c2dx58−u
2π
(c3 logN
d+ c4
)+ d.
Proof. For Re s > 1, the derivative of (26) yields∑p
log p
∞∑m=1
χ(pm)p−ms = −L′(s, χ)
L(s, χ). (31)
Substitute y = pm/x in the classical Cahen–Mellin integral (see [8])
e−y =1
2πi
∫ 2+i∞
2−i∞y−sΓ(s) ds, for y > 0,
EXPLICIT MERTENS’ THEOREMS FOR NUMBER FIELDS 13
and obtain
e−pm/x =
1
2πi
∫ 2+i∞
2−i∞p−msxsΓ(s) ds for x > 0.
For 1 ≤ u ≤ 32 and x > 0, it follows from (31) that
∑p
log p
∞∑n=1
χ(pm)p−mue−pm/x
=∑p
log p
∞∑m=1
χ(pm)p−mu(
1
2πi
∫ 2+i∞
2−i∞p−msxsΓ(s) ds
)
=1
2πi
∫ 2+i∞
2−i∞
(∑p
log p
∞∑m=1
χ(pm)p−m(s+u)
)xsΓ(s) ds
= − 1
2πi
∫ 2+i∞
2−i∞
L′(s+ u, χ)
L(s+ u, χ)xsΓ(s) ds.
Shift the integration contour of integration to the vertical line Re s = 58 −u . Since
Γ has a simple pole with residue 1 at s = 0 and 58 − u < 0, we pick up the residue
Ress=0
(L′(s+ u, χ)
L(s+ u, χ)xsΓ(s)
)=L′(u, χ)
L(u, χ),
and obtain∑p
log p
∞∑m=1
χ(pm)
pmuepm/x+L′(u, χ)
L(u, χ)= − 1
2πi
∫ 58−u+i∞
58−u−i∞
L′(s+ u, χ)
L(s+ u, χ)xsΓ(s) ds. (32)
Since 58 − u ∈ [− 7
8 ,−38 ], we estimate integral on the right-hand side of (32) with
Lemmas 4 and 5:∣∣∣∣∣∫ 5
8−u+i∞
58−u−i∞
L′(s+ u, χ)
L(s+ u, χ)xsΓ(s) ds
∣∣∣∣∣ ≤∫ ∞−∞
∣∣∣∣L′( 58 + it, χ)
L( 58 + it, χ)
∣∣∣∣x 58−u|Γ( 5
8 − u+ it)|dt
≤ c2dx58−u
∫ ∞−∞|Γ( 5
8 − u+ it)| log(N
1d (|t|+ 4)
)dt
≤ c2dx58−u
(c3 logN
d+ c4
).
It follows from the triangle inequality and (32) that∣∣∣∣∣∑p
log p
∞∑m=1
χ(pm)
pmuepm/x+L′(u, χ)
L(u, χ)
∣∣∣∣∣≤
∣∣∣∣∣∑p
log pχ(p)
puep/x+L′(u, χ)
L(u, χ)
∣∣∣∣∣+
∣∣∣∣∣∑p
log p
∞∑m=2
χ(pm)
pmuepm/x
∣∣∣∣∣≤ c2dx
58−u
2π
(c3 logN
d+ c4
)+
∣∣∣∣∣∑p
log p
∞∑m=2
χ(pm)
pmuepm/x
∣∣∣∣∣ . (33)
14 STEPHAN RAMON GARCIA AND ETHAN SIMPSON LEE
Finally, we bound the sum over m ≥ 2 in (33) using |χ(pm)| ≤ d, the inequality
1 ≤ u ≤ 32 , and numerical summation:1∣∣∣∣∑
p
log p
∞∑m=2
χ(pm)
pmuepm/x
∣∣∣∣ ≤ d∑p
log p
∞∑m=2
p−mu
≤ d∑p
log p
pu(pu − 1)
≤ d∑p
log p
p(p− 1)
< d. �
The next step in the proof of Theorem B is the following lemma, which is an ex-
plicit version of the argument at the top of [3, p. 113]. In what follows, the notation
f(t) = O?(g(t)) means that |f(t)| ≤ |g(t)| for all values of t under consideration.
Lemma 7. Let L(s, χ) be an entire Artin L-function that satisfies GRH, in which
χ has degree d and conductor N . For x > 1,∣∣∣∣∣ logL(1, χ)−∑p
χ(p)p−1e−p/x
∣∣∣∣∣ ≤ 5
2d+
c5 logN + c6d
x3/8 log x,
in which
c5 =c2c32π
< 25.5261 and c6 =c2c42π
< 38.9464.
Proof. Fix x > 1. We begin by integrating the expressions in the inequality from
Lemma 6 over u ∈ [1, 32 ] and obtain∫ 32
1
∑p
(log p)χ(p)p−ue−p/x du =∑p
(log p)χ(p)e−p/x∫ 3
2
1
p−u du
=∑p
χ(p)e−p/x√p− 1
p3/2
=∑p
χ(p)p−1e−p/x −∑p
χ(p)p−3/2e−p/x
=∑p
χ(p)p−1e−p/x +O?(d∑p
1
p3/2
)=∑p
χ(p)p−1e−p/x +O?(d)
by numerical summation.2 The initial interchange of integral and summation is
permissible by uniform convergence. Using the estimate (27), we obtain∫ 32
1
L′(u, χ)
L(u, χ)du = logL( 3
2 , χ)− logL(1, χ) = − logL(1, χ) +O?(d log ζ( 32 ))
= − logL(1, χ) +O?(d),
1Mathematica gives∑
plog p
p(p−1)≈ 0.755. A crude upper bound is log 2
2+∑∞
n=1log(2n+1)(2n+1)(2n)
< 0.86.2Mathematica’s built-in PrimeZetaP command or numerical summation give the upper bound 0.85.
EXPLICIT MERTENS’ THEOREMS FOR NUMBER FIELDS 15
since log ζ( 32 ) ≈ 0.9603 < 1. Integrate the upper bound in Lemma 6 and obtain∫ 3/2
1
[c2dx
58−u
2π
(c3 logN
d+ c4
)+ d
]du =
c2d
2π
(c3 logN
d+ c4
)( √x− 1
x7/8 log x
)+d
2
≤ c2d
2π
(c3 logN
d+ c4
)x4/8
x7/8 log x+d
2
=
(c2c32π
logN +c2c4d
2π
)1
x3/8 log x+d
2
=c5 logN + c6d
x3/8 log x+d
2.
Lemma 6 says that∑p
(log p)χ(p)p−ue−p/x = −L′(u, χ)
L(u, χ)+O∗
(c2dx
58−u
2π
(c3 logN
d+ c4
)+ d
).
Integrate this over u ∈ [1, 32 ], use the preceding computations, and obtain∑p
χ(p)p−1e−p/x +O?(d) = L(1, χ) +O∗ (d) +O∗(c5 logN + c6d
x3/8 log x+d
2
),
which is equivalent to the desired result. �
The sum that appears in the previous lemma involves the expression e−p/x. Since
e−p/x = 1 + O(1/x), one might seek to replace e−p/x with 1 if p/x is sufficiently
small (with all constants explicit, of course). This is the content of the next lemma.
Lemma 8. Let L(s, χ) be an entire Artin L-function that satisfies GRH, in which
χ has degree d and conductor N . For x = (logN)3 and y = x1/6 = (logN)1/2,∣∣∣∣∣∑p
χ(p)p−1e−p/x −∑p≤y
χ(p)p−1
∣∣∣∣∣ < 7.81469 d.
Proof. Let x = (logN)3 and y = x1/6 = (logN)1/2. Then∑p
χ(p)p−1e−p/x =∑p≤y
χ(p)p−1 +∑p≤y
χ(p)p−1(e−p/x − 1)︸ ︷︷ ︸I1
+∑
y<p≤x2
χ(p)p−1e−p/x
︸ ︷︷ ︸I2
+∑x2<p
χ(p)p−1e−p/x
︸ ︷︷ ︸I3
.
Bounding I1. Since p ≤ y < x, we may use t = p/x in the inequality
|1− et| ≤ (e− 1)|t|, for |t| < 1, (34)
which follows from the power-series representation of the exponential function and
the triangle inequality. Observe that I1 = 0 if√
logN = y < 2; that is, if N < e4 ≈54.6. For N ≥ 55, we have
|I1| =∣∣∣∑p≤y
χ(p)p−1(e−p/x − 1)∣∣∣ ≤ ∑
p≤y
|χ(p)| |e−p/x − 1|
p
16 STEPHAN RAMON GARCIA AND ETHAN SIMPSON LEE
= d∑p≤y
1− e−p/x
p≤ (e− 1)d
∑p≤y
1
p· px
(by (34))
= (e− 1)d∑p≤y
1
(logN)3≤ (e− 1)d
(logN)52
≤(
e− 1
(log 55)52
)d < 0.05346 d.
Bounding I2. If N = 3, then (y, x2] ≈ (1.048, 1.326] and hence I2 = 0. Conse-
quently, we assume that N ≥ 4. Apply (14) with K = Q and obtain∣∣∣∣∑p≤t
1
p− log log t−M
∣∣∣∣ ≤ 1√t
(3 log t
8π+
18
log t+
5
4π
)︸ ︷︷ ︸
m(t)
, for t ≥ 2, (35)
in which M ≈ 0.2615 is the Meissel–Mertens constant. To see that m(t) is decreas-
ing for t ≥ 2, observe that m(2) > 18.7, m′(2) > −17.9, and
m′(t) =−3(log t)3 − 4(log t)2 − 144π log t− 288π
16πt3/2(log t)2,
has only a single real zero (at t ≈ 0.137648).
For 4 ≤ N ≤ 1012, numerical summation provides3
|I2| =∣∣∣∣ ∑y<p≤x2
χ(p)p−1e−p/x∣∣∣∣ ≤ d ∑
p≤(log(1012))6p−1e−p/x < 2.49045d. (36)
For N > 1012, (35) yields
|I2| =∣∣∣∣ ∑y<p≤x2
χ(p)p−1e−p/x∣∣∣∣ ≤ d ∑
y<p≤x2
p−1e−p/x ≤ d∑
y<p≤x2
p−1
≤ d( ∑p≤x2
p−1 −∑p≤y
p−1)
≤ d((
log log(x2) +M +O?(m(x2)))−(
log log y +M +O?(m(y))))
≤ d(
log
(log(x2)
log y
)+m(x2) +m(y)
)= d
(log
(6 log logN12 log logN
)+m(x2) +m(y)
)= d(
log 12 +m(x2) +m(y))
≤ d(
log 12 +m((log(1012))6
)+m
(√log(1012)
))< 7.47604d.
It follows that |I2| < 7.47604d for N ≥ 3.
3The cutoff 1012 is chosen so that (36) is amenable to computation. In Mathematica, run
Sum[Exp[-p/x]/p, {p, Prime@Range@PrimePi@r}] with x = Log[10^12]^3 and r = x^2.
EXPLICIT MERTENS’ THEOREMS FOR NUMBER FIELDS 17
Bounding I3. We use a geometric series argument. Observe that
|I3| =∣∣∣ ∑x2<p
χ(p)p−1e−p/x∣∣∣ ≤ d ∑
x2<p
p−1e−p/x
≤ d
x2
∑x2<p
e−p/x ≤ d
x2
∞∑k=bx2c
(e−1/x)k
=d
x2(e−1/x)bx
2c
1− e−1/x≤ d(e−1/x)x
2
x2(1− e−1/x)
≤ de−x
x2(1− e−1/x)≤ de−(log 3)3
(log 3)6(1− e−1/(log 3)3)
< 0.28519 d,
since the expression that multiplies d in the penultimate line is decreasing in x and
is thus maximized by x = (log 3)3 because N ≥ 3. Putting this all together we have∣∣∣∣∣∑p
χ(p)p−1e−p/x −∑p≤y
χ(p)p−1
∣∣∣∣∣ < 7.81469 d. �
The numerical constant that appears in the statement of the previous lemma
can be given in closed form by examining the constants that bound I1, I2, and I3in the proof. However, this does not seem worthwhile to pursue.
Remark 9. Under the Riemann Hypothesis, Schoenfeld proved [29, Cor. 2]:∣∣∣∣∣∑p≤t
1
p− log log t−M
∣∣∣∣∣ ≤ 3 log t+ 4
8π√t
, for t ≥ 13.5.
For large t, this is better than our estimate (35). However, the range of validity is
problematic since we require estimates on∑p≤y p
−1, in which y =√
logN . Since√logN ≥ 13.5 requires N ≥ 1.413× 1079, we do not use Schoenfeld’s estimate and
instead revert to our own (35) since it is valid for t ≥ 2.
We are now in a position to complete the proof of Theorem B. With x = (logN)3,
Lemmas 7 and 8 ensure that∣∣∣∣ logL(1, χ)−∑
p≤(logN)1/2
χ(p)
p
∣∣∣∣ ≤ 5
2d+
c5 logN + c6d
x3/8 log x+ 7.81469 d
≤ c5 logN
x3/8 log x+
c6d
x3/8 log x+ 10.3147 d
=c5
3(logN)1/8 log logN+
c6d
3(logN)9/8 log logN+ 10.3147 d
<c5
3(log 3)1/8 log log 3+
c6d
3(log 3)9/8 log log 3+ 10.3147 d
< 89.4146 + 134.494 d. �
18 STEPHAN RAMON GARCIA AND ETHAN SIMPSON LEE
3.4. Proof of (K). In this section, we establish a particular application of Theorem
B. First note that κQ = 1 since the Riemann zeta-function has a simple pole at
s = 1 with residue 1. Thus, we may assume that nK ≥ 2.
Let L(s, χ) denote the Artin L-function from (20) and recall that its conductor is
N = ∆K and χ is the character of a certain (nK−1)-dimensional representation. Let
Cd = 89.5+134.5 d denote the bound on the right-hand side of (5); here d = nK−1.
Since the sums below contain no summands unless N ≥ e4 ≈ 54.6, we may assume
that N ≥ 55. By (22), (35), and Theorem B it follows that
logL(1, χ) =∑
p≤(logN)12
χ(p)
p+O?(Cd)
≤ d∑
p≤(logN)12
1
p+ Cd
≤ d log(12 log logN
)+Md+ dm
(√log 55
)+ Cd
≤ d(
log log logN + log 12 +M +m
(√log 55
))+ Cd
≤ (nK − 1) log log logN + 152.74nK − 63.23.
Since L(1, χ) > 0, it follows that
L(1, χ) ≤ (log logN)nK−1e152.74nK−63.23.
For N ≥ 55, (22), (35), and Theorem B provide
logL(1, χ) ≥ −∑
p≤(logN)12
1
p+O?(Cd)
≥ − log log(√
logN)−M −m(√
logN)− Cd≥ − log log logN − log 1
2 −M −m(√
log 55)− Cd> − log log logN − 134.5nK + 26.76.
Thus,
L(1, χ) ≥ 1
e134.5nK−26.76 log logN.
Since (21) ensures that κK = L(1, χ), this completes the proof of (K). �
References
1. P. J. Cho and H. H. Kim, Extreme residues of Dedekind zeta functions, Math. Proc. Cambridge
Philos. Soc. 163 (2017), no. 2, 369–380. MR 3682635
2. C. J. de la Vallee Poussin, La fonction ζ(s) de Riemann et les nombres premiers en general,
Ann. Soc. Sci. Bruxelles Ser. I 20 (1896), 183–256.
3. W. Duke, Extreme values of Artin L-functions and class numbers, Compositio Math. 136
(2003), no. 1, 103–115. MR 1966783
4. S. R. Garcia and E. S. Lee, Unconditional explicit Mertens’ theorems for number fields and
Dedekind zeta residue bounds, submitted. https://arxiv.org/abs/2007.10313.
5. L. Grenie and G. Molteni, Explicit versions of the prime ideal theorem for Dedekind zeta
functions under GRH, Math. Comp. 85 (2016), no. 298, 889–906. MR 3434887
6. , An explicit Chebotarev density theorem under GRH, J. Number Theory 200 (2019),
441–485. MR 3944447
EXPLICIT MERTENS’ THEOREMS FOR NUMBER FIELDS 19
7. J. Hadamard, Sur la distribution des zeros de la fonction ζ(s) et ses consequences
arithmetiques, Bull. Soc. Math. France 24 (1896), 199–220.
8. G. H. Hardy and J. E. Littlewood, Contributions to the theory of the riemann zeta-function
and the theory of the distribution of primes, Acta Math. 41 (1916), 119–196.
9. A. E. Ingham, The Distribution of Prime Numbers, Cambridge Mathematical Library, Cam-
bridge University Press, Cambridge, 1990, Reprint of the 1932 original, With a foreword by
R. C. Vaughan. MR 1074573
10. H. Iwaniec, E. Kowalski, and American Mathematical Society, Analytic number theory, Amer-
ican Mathematical Society Colloquium Publications, American Mathematical Society, 2004.
11. G. Kresin and V. Maz′ya, Sharp real-part theorems, Lecture Notes in Mathematics, vol. 1903,
Springer, Berlin, 2007, A unified approach, Translated from the Russian and edited by T.
Shaposhnikova. MR 2298774
12. J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem,
Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham,
Durham, 1975), 1977, pp. 409–464. MR 0447191
13. E. Landau, Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes, Math. Ann.
56 (1903), no. 4, 645–670. MR 1511191
14. S. Lang, Algebraic number theory, second ed., Springer-Verlag, 2000.
15. P. Lebacque, Generalised Mertens and Brauer-Siegel theorems, Acta Arith. 130 (2007), no. 4,
333–350. MR 2365709
16. J. E. Littlewood, On the Class-Number of the Corpus P (√− k), Proc. London Math. Soc.
(2) 27 (1928), no. 5, 358–372. MR 1575396
17. S. Louboutin, Explicit bounds for residues of Dedekind zeta functions, values of L-functions at
s = 1, and relative class numbers, J. Number Theory 85 (2000), no. 2, 263–282. MR 1802716
18. , Explicit upper bounds for residues of Dedekind zeta functions and values of L-
functions at s = 1, and explicit lower bounds for relative class numbers of CM-fields, Canad.
J. Math. 53 (2001), no. 6, 1194–1222. MR 1863848
19. , Explicit lower bounds for residues at s = 1 of Dedekind zeta functions and rel-
ative class numbers of CM-fields, Trans. Amer. Math. Soc. 355 (2003), no. 8, 3079–3098.
MR 1974676
20. S. R. Louboutin, Upper bounds for residues of Dedekind zeta functions and class numbers of
cubic and quartic number fields, Math. Comp. 80 (2011), no. 275, 1813–1822. MR 2785481
21. F. Mertens, Ein Beitrag zur analytischen Zahlentheorie, J. Reine Angew. Math. 78 (1874),
46–62. MR 1579612
22. H. L. Montgomery and R. C. Vaughan, Multiplicative number theory I: Classical theory,
reprint ed., Cambridge Studies in Advanced Mathematics (Book 97), 2012.
23. J. Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fun-
damental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999, Trans-
lated from the 1992 German original and with a note by Norbert Schappacher, With a foreword
by G. Harder. MR 1697859 (2000m:11104)
24. J. Oesterle, Versions effectives du theoreme de Chebotarev sous l’hypothese de Riemann
generalisee, Journees Arithmetiques de Luminy, Societe Mathematique de France, Paris, 1979,
Colloque International du Centre National de la Recherche Scientifique, June 20–24, 1978,
Asterisque No. 61 (1979), pp. 165–167. MR 556662
25. F. W. Olver, D. W. Lozier, R. F. Boisvert, and Charles W. Clark, Nist handbook of mathe-
matical functions, 1st ed., Cambridge University Press, USA, 2010, https://dlmf.nist.gov/
5.6.E9.
26. H. Rademacher, On the Phragmen-Lindelof theorem and some applications, Math. Z 72
(1959/1960), 192–204. MR 0117200
27. M. Rosen, A generalization of Mertens’ theorem, J. Ramanujan Math. Soc. 14 (1999), no. 1,
1–19.
28. J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers,
Illinois J. Math. 6 (1962), 64–94. MR 0137689
29. L. Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II, Math. Comp.
30 (1976), no. 134, 337–360. MR 0457374
20 STEPHAN RAMON GARCIA AND ETHAN SIMPSON LEE
30. J.-P. Serre, Local class field theory, Algebraic Number Theory (Proc. Instructional Conf.,
Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 128–161. MR 0220701
31. H. M. Stark, A complete determination of the complex quadratic fields of class-number one,
Michigan Math. J. 14 (1967), 1–27. MR 0222050
32. B. Winckler, Theoreme de Chebotarev effectif, (2013), https://arxiv.org/abs/1311.5715.
Department of Mathematics, Pomona College, 610 N. College Ave., Claremont, CA
91711
Email address: [email protected]
URL: http://pages.pomona.edu/~sg064747
School of Science, UNSW Canberra at the Australian Defence Force Academy,
Northcott Drive, Campbell, ACT 2612
Email address: [email protected]
URL: https://www.unsw.adfa.edu.au/our-people/mr-ethan-lee