Modular forms and an explicit Chebotarev variant of the Brun-Titchmarsh theorem

Abstract

We prove an explicit Chebotarev variant of the Brun--Titchmarsh theorem. This leads to explicit versions of the best-known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that lim⁡x→∞#{1≤n≤x∣τ(n)≠0}x>1−1.15×10−12,\lim_{x \to \infty} \frac{\#\{1 \leq n \leq x \mid \tau(n) \neq 0\}}{x} > 1-1.15 \times 10^{-12}, where τ(n)\tau(n) is Ramanujan's tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers n such that τ(n)≠0\tau(n) \neq 0.

Full text

MODULAR FORMS AND AN EXPLICIT CHEBOTAREV VARIANT
OF THE BRUN–TITCHMARSH THEOREM
DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
Abstract. We prove an explicit Chebotarev variant of the Brun–Titchmarsh theo-
rem. This leads to explicit versions of the best known unconditional upper bounds
toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal
newforms. In particular, we prove that
lim
x→∞
#{1≤n≤x|τ(n)= 0}
x>1−1.15 ×10−12,
where τ(n) is Ramanujan’s tau-function. This is the first known positive unconditional
lower bound for the proportion of positive integers nsuch that τ(n)= 0.
Contents
1. Introduction 1
2. Preliminaries 7
3. Reduction to the abelian case 13
4. Sums over integral ideals 14
5. Implementing the Selberg sieve 20
6. Brun–Titchmarsh for abelian extensions 22
7. Upper bounds for the Lang–Trotter conjecture 23
8. Proof of Theorems 1.2 and 1.4 33
9. Proof of Theorems 1.3 and 1.5 36
References 38
1. Introduction
Let π(x;q, a) be the number of primes p≤xsuch that p≡a(mod q). The prime
number theorem for primes in arithmetic progressions states that if gcd(a, q) = 1, then
for any N > 0, there exists an (ineffective) constant CN>0 such that if q≤(log x)N,
then π(x;q, a)−Li(x)
φ(q)≪Nxexp(−CNplog x)),(1.1)
where Li(x) is the logarithmic integral
Li(x) = Zx
2
dt
log t∼x
log x.
Date: August 22, 2022.
2020 Mathematics Subject Classification. Primary: 11R44; Secondary: 11N36, 11F30.
1

2 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
Under the generalized Riemann hypothesis (GRH) for Dirichlet L-functions, we can
extend the range to q≤x1/2(log x)−1, and reduce the error term to O(x1/2log x). For
many applications, providing a better range for xis of crucial importance, though this
may require the tradeoff that we work with upper or lower bounds for π(x;q, a) in lieu
of an asymptotic. For instance, the Brun–Titchmarsh theorem in the form proved by
Montgomery and Vaughan [MV73] states that if q < x, then
π(x;q, a)≤2x
φ(q) log(x/q).(1.2)
We consider analogous results over number fields. Let L/F be a Galois extension of
number fields with Galois group G. For each prime ideal pof Fwhich does not ramify
in L, we use the Artin symbol L/F
pto denote the conjugacy class in Gof Frobenius
elements of primes in Llying over p. Given a fixed conjugacy class Cof G, define
πC(x, L/F ):= #npunramified in L, L/F
p=C, NF/Qp≤xo.(1.3)
The Chebotarev density theorem states that
lim
x→∞
πC(x, L/F )
Li(x)=|C|
|G|.(1.4)
The first asymptotic version of this result with an effective error term was obtained by
Lagarias and Odlyzko [LO77]. They showed that if x≥exp(10[L:Q](log DL)2), then
there exist absolute, effectively computable constants c0, c1>0 such that
πC(x, L/F )−|C|
|G|Li(x)≤|C|
|G|Li(xβ0) + c0xexp(−c1[L:Q]−1/2(log x)1/2),(1.5)
where the β0term is only present when the Dedekind zeta function ζL(s) has a Siegel
zero at β0. Lagarias, Montgomery, and Odlyzko [LMO79] improved the range at the
cost of only proving an upper bound for πC(x, L/F ). They showed that there exist
absolute, effectively computable constants c2, c3>0 such that
πC(x, L/F )≤c2|C|
|G|Li(x),log x≥c3(log DL)(log log DL)(log log log e20DL).(1.6)
Refinements to the effective version of the Chebotarev density theorem were later made
by Serre [Ser81] and V. K. Murty [Mur97].
In a series of papers [TZ17,TZ18,TZ19], Thorner and Zaman improved the previous
results on the asymptotic form and upper and lower bounds for the Chebotarev density
theorem. Here we focus on their analogue of the Brun–Titchmarsh theorem (1.2). Let
A⊆Gbe an abelian subgroup such that C∩Ais nonempty, and let Kbe the fixed
field of A. Set nK= [K:Q]. Let b
Abe the dual group, fχthe conductor of χ∈b
A, and
define
Q=Q(L/K):= max
χ∈b
A
NK/Qfχ.(1.7)

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 3
Thorner and Zaman [TZ18, Theorem 1.1] proved that there exists an absolute, effec-
tively computable constant c4>0 such that if
x≥c4D164
KQ123 +D55
KQ87n68nK
K+D2
KQ2n14,000nK
K,(1.8)
then the bound on πC(x, L/F ) in (1.6) holds. We have that DKQ ≤ DLby the
conductor-discriminant formula, and by Minkowski’s bound, there exists an absolute,
effectively computable constant c5>0 such that nK≤c5log DK≤c5log DL. Thus,
there exists an absolute, effectively computable constant c6>0 such that (1.8) holds
when log x≥c6(log DL)(log log DL), so (1.8) is a uniform improvement over the range
in (1.6). There also exists an absolute, effectively computable constant c7>0 such
that for most number fields K,nK≤c7(log DK)/log log DK. In this case, there exists
an absolute, effectively computable constant c8>0 such that (1.8) holds when log x≥
c8log DL.
Our main result is a completely explicit Chebotarev variant of the Brun–Titchmarsh
theorem with a range of xthat improves upon (1.8).
Theorem 1.1. Let L/F be a Galois extension of number fields with Galois group G,
and let C⊂Gbe a conjugacy class. Let Abe an abelian subgroup of Gsuch that C∩A
is nonempty, and let Kbe the fixed field of A. If
x≥e92nK+36(DKQ)8.4n4.2nK
K,(1.9)
then
πC(x, L/F )≤11.3|C|
|G|
x
log x.(1.10)
Remark 1.1. Note that x
log x≤Li(x) for all x≥10.
Remark 1.2. In the case where F=K=Qand L=Q(e2π i/q), we recover a weakened
leading constant in the classical Brun–Titchmarsh theorem, valid in the range x≥
e128q8.4.
To prove Theorem 1.1, we use the Selberg sieve over number fields as in [Wei83]. In
order to implement this strategy, we obtain estimates for the number of integral ideals
in Kwith norm less than xsatisfying certain congruence conditions. We prove these
by bounding contour integrals of Hecke L-functions smoothed by a test function. This
input may alternatively be treated by translation to a lattice point-counting argument,
as in work of Debaene [Deb19], who obtained a similar result with better exponents
of DKand Qbut worse dependence on nK. On the other hand, Debaene only works
with the base field F=Q, whereas the applications which motivate our work involve
results over a general base field F. Moreover, the nK-dependence is crucial for our
applications.
We apply Theorem 1.1 to study values of the Fourier coefficients of modular forms.
For Im(z)>0, let
f(z) = ∞
X
n=1
af(n)e2πinz ∈ Snew
k(Γ0(N))

4 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
be a normalized cusp form of even weight k≥2 which is a newform of level N=Nf
with trivial nebentypus and without complex multiplication. Fix a∈Z, and define
πf(x, a):= #{p≤x, p ∤Nf, af(p) = a}.(1.11)
One of the most celebrated such examples is the function
∆(z) = e2πiz ∞
Y
n=1
(1 −e2πinz )24 =∞
X
n=1
τ(n)e2πinz (1.12)
=e2πiz −24e4πiz + 252e6πiz −1472e8πiz + 4830e10πiz − · · · ,(1.13)
where τ(n) is Ramanujan’s tau-function. Lehmer [Leh47] pondered whether τ(n)= 0
for all n≥1, and showed that proving τ(p)= 0 for all primes pwould imply this. The
question remains open, with computations by Bosman [EC11] verifying that τ(n)= 0
for n≤2×1019. Closely related to Lehmer’s speculation is the following conjecture of
Atkin and Serre [Ser76].
Conjecture (Atkin–Serre).Let f∈ Snew
k(Γ0(N)) be a non-CM newform with trivial
nebentypus. For all ϵ > 0, there exist constants cf(ϵ), c′
f(ϵ)>0such that if p≥cf(ϵ),
then |af(p)| ≥ c′
f(ϵ)p(k−3)/2−ϵ.
In particular, the truth of this conjecture would imply for all a∈Zthat τ(p) = afor
only finitely many p. In light of this conjecture, it is natural to ask for bounds on the
function π∆(x, a). Serre [Ser81] used refined versions of (1.5) and (1.6) to show that
for all δ < 1/4, there exists an effectively computable constant c(δ)>0 such that
π∆(x, a)≤c(δ)x
(log x)1+δ, x ≥3.(1.14)
Following subsequent improvements by Wan [Wan90] and V. K. Murty [Mur97], Thorner
and Zaman [TZ18] proved that there exists an absolute, effectively computable constant
c9>0 such that
π∆(x, a)≤c9
x(log log x)2
(log x)2, x ≥16.(1.15)
Under GRH for symmetric power L-functions, Rouse and Thorner [RT17] proved
a stronger bound using arguments based on the Sato–Tate conjecture. They proved,
under GRH for symmetric power L-functions, that there exists an absolute, effectively
computable constant c10 >0 such that for all a∈Z, there exists an effectively com-
putable constant c(a)>0 such that
π∆(x, a)≤c10
x3/4
√log x, x ≥c(a).(1.16)
In the case a= 0, Zywina [Zyw15] matched this bound under GRH for Hecke L-
functions using arguments based on the Chebotarev density theorem.
We make the result (1.15) explicit by proving the following.
Theorem 1.2. For all a∈Z, we have that
π∆(x, a)≤4627x(log log x)2
(log x)2, x ≥ee16 .

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 5
When a= 0, this bound may be strengthened to
π∆(x, 0) ≤(3.01 ×10−10)x(log log x)2
(log x)2, x ≥ee16 .
For the weight 2 newform
fE(z) = ∞
X
n=1
aE(n)e2πinz ∈ Snew
2(Γ(NE))
associated to a non-CM elliptic curve E/Qof conductor NE, Lang and Trotter [LT76]
conjectured that there exists a constant cE,a ≥0 such that
πE(x, a):=πfE(x, a)∼cE ,a
√x
log x.(1.17)
When a= 0, Elkies [Elk87,Elk91] and Fouvry and M. R. Murty [FM96] showed that
for all ϵ > 0, there exist positive constants c(E, ϵ), c′(E, ϵ), and c(E) such that if
x≥c(E, ϵ), then
c′(E, ϵ)log log log x
(log log log log x)1+ϵ≤πE(x, a)≤c(E)x3/4.(1.18)
In particular, aE(p) = 0 infinitely often. In the general case where a∈Zis fixed,
Serre [Ser81] proved that for all δ < 1/4, there exists an effectively computable constant
c(NE, δ)>0 such that
πE(x, a)≤c(NE, δ)x
(log x)1+δ, x ≥3.(1.19)
After subsequent improvements by Wan [Wan90] and V. K. Murty [Mur97], Thorner and
Zaman [TZ18] proved that there exists an effectively computable constant c(NE)>0
such that
πE(x, a)≤c(NE)x(log log x)2
(log x)2, x ≥3.(1.20)
For an example, consider the elliptic curve
E:y2−y=x3−x2.(1.21)
This elliptic curve has conductor 11, the least conductor of any elliptic curve over Q.
Its associated modular form is given by
fE(z) = η2(z)η2(11z), η(z):=eπiz/12 Y
n≥1
(1 −e2πinz ).(1.22)
We prove the following theorem.
Theorem 1.3. Let Ebe the elliptic curve given by (1.21)and a∈Z. Then,
πE(x, a)≤631x(log log x)2
(log x)2, x ≥ee13 .(1.23)
Remark 1.3. Analogues of Theorems 1.2 and 1.3 can be obtained for a broad class of
other newforms, see Remark 7.1.

6 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
Suitably strong bounds for the quantity πf(x, 0) as in (1.11) can be used to study
the proportion of integers n≥1 such that af(n)= 0. Define
Df:= lim
x→∞
#{1≤n≤x|af(n)= 0}
x.
For f= ∆, Serre [Ser81, p. 379] proved that
D∆=Y
τ(p)=0 1−1
p+ 1.(1.24)
As a consequence of (1.14), this product converges absolutely. Rouse and Thorner [RT17]
showed that D∆>1−1.54 ×10−13 under the generalized Riemann hypothesis for the
symmetric power L-functions of ∆. Here, we prove the first unconditional positive lower
bound for D∆.
Theorem 1.4. We have
D∆>1−1.15 ×10−12.(1.25)
For the cusp form fEassociated to the elliptic curve given in (1.21), Serre [Ser81,
p. 379] proved that
DfE=14
15 Y
aE(p)=0 1−1
p+ 1.(1.26)
As a consequence of (1.19), the product converges absolutely. Serre showed that DfE<
0.847 and conjectured a lower bound of 0.845. Rouse and Thorner [RT17] proved that
DfE>0.8306 under the generalized Riemann hypothesis for the symmetric power L-
functions associated to E. We prove the first unconditional positive lower bound for
DfE.
Theorem 1.5. Let Ebe given by (1.21). We have that
DfE>0.004769.
The density D∆naturally arises in a central limit theorem for τ(n) proved by Luca,
Radziwi l l, and Shparlinski [LRS19]. Denote by d(n) the number of divisors of n, and
recall, by Deligne’s proof of the Weil conjectures, that for all n≥1, the function τ(n)
satisfies the bound
|τ(n)| ≤ d(n)n11/2.(1.27)
Consequently, it is natural to consider the distribution of a “normalized” tau-function
τ(n)/n11/2at the integers n≥1. For instance, the Sato–Tate conjecture implies that
|τ(p)/p11/2|stays arbitrarily near the value 2 for a positive proportion of primes p. In
fact, an application of Theorem 1 of [LRS19] demonstrates that for all ϵ > 0, there
exists a density one subset Sϵof the integers such that if n∈Sϵ, then
|τ(n)|/n11/2≤(log n)−1
2+ϵ.
This implies that for almost all n≥1, Deligne’s bound (1.27) for τ(n)/n11/2is not
sharp.

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 7
Moreover, Corollary 5 of [LRS19] shows that the exponent −1
2above is optimal, in
the sense that for fixed v∈R, the central limit theorem
lim
x→∞
#n1≤n≤xlog |τ(n)/n11/2|+(log log n)/2
q(1
2+π2
12 ) log log n≥vo
#{1≤n≤x|τ(n)= 0}=1
√2πZ∞
v
e−u2/2du
holds.
This gives the distribution as a proportion of the integers in the support of τ(n), but
by substituting the upper bound D∆≤1 and our lower bound for D∆as in Theorem 1.4,
we establish upper and lower limits for the distribution as a proportion of all integers
in the style of the Erd˝os–Kac theorem.
Corollary 1.6. For fixed v∈R, we have
1
√2πZ∞
v
e−u2/2du ≥lim
x→∞
1
x·#1≤n≤x:log |τ(n)/n11/2|+1
2log log n
q(1
2+π2
12 ) log log n≥v
>(1 −1.15 ×10−12)1
√2πZ∞
v
e−u2/2du.
Outline of the paper. In Section 2, we introduce notation and prove some preliminary
technical lemmas, including estimates for a certain test function and bounds for Hecke
L-functions in the critical strip. In Section 3, we show that Theorem 1.1 follows from
only considering abelian extensions of number fields. In Section 4, we use smoothing
arguments to compute estimates for certain sums over integral ideals. We apply these
estimates in Section 5, where we use the Selberg sieve over number fields to develop
an upper bound for πC(x, L/K ) for abelian extensions. In Section 6we bound error
terms arising from the Selberg sieve by choosing xin a large enough range relative
to parameters depending on L/K, thereby yielding our Brun–Titchmarsh theorem for
abelian extensions.
In Section 7, we deduce an upper bound for πf(x, a) from upper bounds for πC(x, L/K )
for suitably chosen extensions L/K arising from ℓ-adic Galois representations attached
to f. In Section 8we make the results in Section 7explicit for τ(n), proving Theo-
rem 1.2 and Theorem 1.4. Finally, in Section 9we prove Theorems 1.3 and 1.5 for the
elliptic curve (1.21).
Acknowledgments. The authors would like to thank Ken Ono for his valuable sugges-
tions. They are grateful for the support of grants from the National Science Foundation
(DMS-2002265, DMS-2055118, DMS-2147273), the National Security Agency (H98230-
22-1-0020), and the Templeton World Charity Foundation. This research was conducted
as part of the 2022 Research Experiences for Undergraduates at the University of Vir-
ginia.
2. Preliminaries
Throughout this paper s=σ+it is a complex variable.

8 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
2.1. Notation. In this section we establish the notation utilized in the proof of Theo-
rem 1.1. Subsequently Kwill be a number field and La finite extension of K. We use
the following notation:
• OKis the ring of integers of K.
•nK= [K:Q] is the degree of K/Q.
•DK=|disc(K/Q)|.
•NK/Qis the absolute norm of K. We write N = NK/Qwhen there is no ambiguity
about the field.
•ζK(s) is the Dedekind zeta function of K.
•κKis the value of the residue of the pole of ζK(s) at s= 1.
•hKis the class number of K.
•pdenotes a prime ideal of K.
•adenotes an integral ideal of K.
•ω(a) is the number of distinct prime ideals dividing a.
•φK(a) = (Na)Qp|a(1 −Np−1) is the Euler phi-function associated to K.
Suppose that L/K is a Galois extension of number fields, and let Gbe its Galois
group. Recall the Artin symbol (L/K
p), also denoted Frobp, associated to a prime ideal
pof Kunramified in L; it is the conjugacy class in Gconsisting of Frobenius automor-
phisms corresponding to the prime ideals Plying over p.
Let L/K be abelian. Then, each conjugacy class of Gis a singleton. Thus, the Artin
symbol maps unramified prime ideals of Kto elements of G. For man integral ideal of
K, let I(m) denote the group of fractional ideals relatively prime to m, and let Pmbe
the subgroup of principal ideals (α) with α∈K×totally positive and α≡1 (mod m).
Then the ray class group modulo mis defined to be the finite group I(m)/Pm.
If mis divisible by all ramified primes in L/K, then the Artin symbol extends mul-
tiplicatively to a surjective group homomorphism called the Artin map
FL/K :I(m)→Gal(L/K).
By Artin reciprocity, there exists an integral ideal msuch that
H= ker(FL/K)
contains Pm. In this case, FL/K induces an isomorphism
I(m)/H ∼
=Gal(L/K).(2.1)
Moreover, there exists a unique integral ideal fL/K , called the Artin conductor of L/K ,
dividing all mwith this property. Thus [I(fL/K ) : H] = [L:K].
Let Cbe one conjugacy class of G. While our goal is to bound the quantity
πC(x, L/K), the above discussion implies that this problem is equivalent to the problem
of counting prime ideals in a given coset of Hin the ray class group modulo fL/K . In par-
ticular, our conjugacy class C⊂Gis identified with a unique coset aH ∈I(fL/K )/H, so
the prime ideals pwith Frobp=Care precisely the prime ideals p∈aH. In summary,
we have that
πC(x, L/K) = πaH (x, L/K),(2.2)

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 9
where πC(x, L/K) is given by (1.3) and
πaH (x, L/K):= #npunramified in L/K, p∈aH, NK/Q(p)≤xo.(2.3)
We recall the basic theory of ray class characters and their associated L-functions.
Let mbe an integral ideal. A ray class character χmodulo mis defined to be a
character of the group I(m)/Pm, with the convention that χ(a) = 0 if aand mare
not coprime. Note that we will often view χas acting on an ideal aitself rather than
its ideal class aPmin the ray class group. If mdivides nthen the inclusion induces a
map I(n)/Pn→I(m)/Pm; composing a character χmod mwith this map induces a
character χ′mod n. A character is primitive if it cannot be induced, except by itself.
Given some χmod m, there is a unique primitive character eχmod fχwhich induces χ.
The integral ideal fχis the conductor of χ.
If m=fL/K and H= ker(FL/K), then we define the following quantity associated to
the extension L/K,
Q=Q(L/K):= max
χ(H)=1 NK/Qfχ,
where the maximum is over characters χsuch that χ(H) = 1; by (2.1), these correspond
precisely to the irreducible characters of G.
Given an integral ideal mand a ray class character χmodulo m, the Hecke L-function
associated to χis given by the following Dirichlet series and Euler product:
L(s, χ) = X
a
χ(a)
Nas=Y
p1−χ(p)
Nps−1, σ > 1,
with the sum over all integral ideals of Kand the product over all prime ideals of K.
For the purpose of obtaining explicit numerical constants, careful bounds on Hecke
L-functions along vertical lines near the critical strip will also be of importance. These
are derived by application of the Phragm´en–Lindel¨of principle for complex analytic
functions in Lemma 2.1, taking as input the functional equation for the the Hecke
L-functions and asymptotics for related gamma factors. First we define the quantities
Dχ=DKNK/Qfχ.
δχ0(χ) = (1χ=χ0
0 otherwise
Let r1be the number of real places of Kand 2r2the number of complex places of K.
If χis primitive, then let 0 ≤µχ≤r1be the number of real places at which χramifies,
and set
Aχ= 2−r2πnK/2D1/2
χ,
Γχ(s)=Γs
2r1−µχΓs+ 1
2µχΓ(s)r2.
The completed Hecke L-function is given by
Λ(s, χ) = As
χΓχ(s)L(s, χ).

10 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
For χa primitive character, Λ(s, χ) satisfies the functional equation
Λ(1 −s, χ) = W(χ)Λ(s, χ),
with the root number W(χ) a complex constant of modulus 1.
2.2. Preliminary estimates. We begin by establishing explicit bounds for the Hecke
L-function of a ray class character in the critical strip.
Lemma 2.1. Let mbe an integral ideal of K, and let χbe a primitive ray class character
modulo m. If δ > 0and −δ < σ < 1, then
|L(s, χ)| ≤ δ+ 1
1−σδχ0(χ)(1 + δ−1)nKDχ
(2π)nK|s+ 1|nK(1+δ−σ)/2.(2.4)
Proof. Expanding the functional equation into unsymmetric form and taking absolute
values yields
|L(s, χ)|=|L(1 −s, χ)|Γχ(1 −s)
Γχ(s)A1−2s
χ.
Using [Rad60, Lemmas 1, 2, 3], we bound the gamma factors at s=−δ+it with
δ∈(0,1/2):
Γχ(1 −s)
Γχ(s)=Γ(1−s
2)
Γ(s
2)r1−µχΓ(1
2+1−s
2)
Γ(1
2+s
2)µχΓ(1 −s)
Γ(s)r2
≤1
2|s+ 1|(1/2+δ)(r1−µχ)1
2|s+ 1/2|(1/2+δ)µχ|s+ 1|(1+2δ)r2
≤2−(1/2+δ)r1|s+ 1|(1/2+δ)nK
where we use that r1+ 2r2=nK. Therefore, if δ∈(0,1/2), then
|L(−δ+it, χ)| ≤ |L(1 + δ+it, χ)|Dχ
(2π)nK|s+ 1|nK(1/2+δ).
By the estimate
|L(s, χ)| ≤ ζK(σ)≤ζ(σ)nK≤1 + 1
σ−1nK, σ > 1,
we also have the bound
|L(1 + δ+it, χ)| ≤ (1 + δ−1)nK.
Now, assume that χis nontrivial, so L(s, χ) is holomorphic in the strip −δ≤σ≤1+ δ.
Then in the strip −δ≤σ≤1 + δ, applying the Phragm´en–Lindel¨of principle gives
|L(s, χ)| ≤ (1 + δ−1)nKDχ
(2π)nK|s+ 1|nK(1+δ−σ)/2.
When χis trivial, we have that L(s, χ) = ζK(s). Since ζK(s) has a pole at s= 1, we
choose any B≥1 and we instead estimate s+B
s−1ζK(s). By a similar argument as above,
in the strip −δ≤σ≤δ+ 1 we have that
|ζK(σ+it)| ≤ δ+ 1
δ−Bs+B
s−1(1 + δ−1)nKDK
(2π)n|s+ 1|nK(1+δ−σ)/2.(2.5)

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 11
Using the bound
s+B
s−1=B+ 1
s−1+ 1≤B+ 1
1−σ+ 1 = B+σ
1−σ,(2.6)
we find that
|ζK(σ+it)| ≤ δ+ 1
1−σ(1 + δ−1)nKDK
(2π)nK|s+ 1|nK(1+δ−σ)/2,(2.7)
since |B+σ|/|δ−B|can be made arbitrarily close to 1 as B→ ∞.□
In order to obtain explicit estimates for πC(x, L/K), we utilize a test function which
approximates the indicator function of the interval [1
2,1].
Lemma 2.2 (Thorner–Zaman [TZ18, Lemma 2.2]).For any x≥3,ϵ∈(0,1
2), and
positive integer ℓ≥1, set
A=ϵ
2ℓlog x.
There exists a real-variable function ϕ(t) = ϕ(t;x, ℓ, ϵ)such that:
(a) 0 ≤ϕ(t)≤1for all t∈R, and ϕ(t)≡1for 1
2≤t≤1.
(b) The support of ϕis contained in the interval [1
2−ϵ
log x,1 + ϵ
log x].
(c) Its Laplace transform F(z) = RRϕ(t)e−zt dt is entire and is given by
F(z) = e−(1+2ℓA)z·1−e(1
2+2ℓA)z
−z1−e2Az
−2Az ℓ.(2.8)
(d) Let s=σ+it ∈C,σ > 0and let αbe any real number satisfying 0≤α≤ℓ.
Then
|F(−slog x)| ≤ eσϵxσ
|s|log x·(1 + x−σ/2)·2ℓ
ϵ|s|α.(2.9)
(e) We have
F(0) = 1
2+ϵ
log xand F(−log x)≤eϵx
log x.(2.10)
(f) If s=σ+it ∈Cand σ≤0, then
|F(−slog x)| ≤ 2xσ/2
|s|log x·ℓ(1 + e−ϵσ)
ϵ|s|ℓ.(2.11)
(g) If s=σ+it with σ < 0and |s|<2ℓ
ϵ, we have that
|F(−slog x)| ≤ 2xσ/2
|s|log x·2
2− |sϵ/ℓ|ℓ.(2.12)
Proof. Parts (a)–(e) are proven in Lemma 2.2 of [TZ18]. Part (f) follows from (2.8).
Part (g) follows from (2.8) and the following identity, which is valid for any complex
|z|<2: 1−ez
−z≤2
2− |z|.(2.13)

12 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
Thorner and Zaman [TZ18] additionally assumed that ϵ∈(0,1
4) in order to obtain
bounds for F(−slog x) on the line σ=−1/2. We have no need for such estimates, so
we can relax this to ϵ∈(0,1
2). □
2.3. Auxiliary results. Assume throughout that L/K is an abelian extension of num-
ber fields. The following lemmas will also be useful.
Lemma 2.3. For all 0< γ ≤1, we have
X
N(a)≤x
1≤(1 + γ−1)nKx1+γ.(2.14)
Proof. See [Wei83, Lemma 1.12(a)]. □
Lemma 2.4. Let a, b > 0. If x≥(ab)a(√2/log(ab)+1), then
x1/a
log x≥b. (2.15)
Proof. By properties of the Lambert W-function [CGH+96], we have that (2.15) holds
when
log x≥ −aW−1−1
ab 
where W−1(z) is the negative branch of the Lambert W-function. By Theorem 1 of
[Cha13], we have that
−aW−1−1
ab ≤ap2 log(ab) + log(ab)≤a(p2/log(ab) + 1) log(ab),
and the lemma follows by taking exponents. □
Next, we bound [L:K] in terms of nK,DKand Qalone.
Lemma 2.5. For all 0< ϵ ≤1,
[L:K]≤2nK(1 + ϵ−1)(nKe)1+ϵ4
e2πnK(1+ϵ)/2D(1+ϵ)/2
KQ.
At ϵ= 1, for instance, this yields
[L:K]≤(14.779)(0.7169)nKDKQ.
Proof. By [Wei83, Lemma 1.16], for Hthe congruence subgroup attached to the Artin
conductor fL/K , we have [L:K] = [I(fL/K ) : H]≤2nKhKQ. By Minkowski’s bound
and [Wei83, Lemma 1.12],
hK≤X
N(a)≤x0
1≤(1 + ϵ−1)nKx1+ϵ
0
for x0=n!
nn(4
π)n/2D1/2
K≤(n
en−1)( 4
π)n/2D1/2
K. Then, an explicit bound is
hK≤(1 + ϵ−1)(nKe)1+ϵ4
e2πnK(1+ϵ)/2D(1+ϵ)/2
K.
Note that for nK≥1 we have nK≤enK−1. At ϵ= 1, note that n2≤(32/3)nfor all
n∈Z≥1.□

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 13
Next, we set m=fL/K and state the following bound for the quantity ω(m).
Lemma 2.6. Let m=fL/K . For every b > 0we have that
ω(m)<2e1+2/bnK+blog DKQ.(2.16)
Proof. See [Wei83, Lemma 1.13(b)]. □
Next, we introduce and bound the following product factor to appear in our treatment
of non-primitive Hecke L-functions.
Lemma 2.7. Let M, b, σ, x > 0. Define
Zm(σ):=Y
p|m
(1 + Np−σ).(2.17)
Let π(x):= #{p≤x, p prime}. We have that
log Zm(σ)≤nKhlog(1 + M−σ)(2e1+2/b −π(M−1)) + X
p<M
log(1 + p−σ)i
+blog(1 + M−σ) log(DKQ).(2.18)
Proof. Given a rational prime p, there are at most nKprime ideals in Kwith norm p.
Thus, the product over all the factors in Zm(σ) with norm at most Mis bounded by
Y
p<M
(1 + p−σ)nK.(2.19)
There are at most ω(m)−nKπ(M−1) prime ideals pdividing msuch that N(p)≥M.
In the product for Zm(σ), the factor corresponding to these primes can be bounded
above by (1 + M−σ), so that the product over all the factors in Zm(σ) with norm at
least Mis bounded by
(1 + M−σ)ω(m)−nKπ(M−1).(2.20)
Multiplying equations (2.19) and (2.20), taking logarithms, and applying Lemma 2.6
completes the proof. □
Note that we could trivially bound Zm(σ) by 2ω(m), but using Lemma 2.7 allows us to
separately bound the contribution from small divisors of mby choosing Moptimally,
which provides sharper bounds later on.
3. Reduction to the abelian case
In this section, we show that Theorem 1.1 follows from the following version of the
Brun–Titchmarsh theorem for abelian extensions, which we prove in Section 6. This
allows us to specialize to abelian extensions in Sections 4and 5, wherein we develop
auxiliary technical results.
Lemma 3.1. Let L/K be an abelian extension of number fields, let Cbe a conjugacy
class (which will necessarily be a singleton) of Gal(L/K), and let πC(x, L/K )be the
prime-counting function given in (1.3). If
x≥e36e92n(DKQ)8.4n4.2n,(3.1)

14 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
then
πC(x, L/K)≤11.29 x
[L:K] log x.(3.2)
We show that Theorem 1.1 follows from this theorem for a general (nonabelian)
Galois extension of number fields L/F . To do this, we change the base field from Fto
an intermediate field Ksuch that L/K is abelian, so that we can apply Lemma 3.1.
The key tool is the following result from [MMS88, Proof of Proposition 3.9].
Lemma 3.2 (Murty–Murty–Saradha [MMS88]).Let L/F be a Galois extension of
number fields with Galois group G, and let C⊂Gbe a conjugacy class. Let Abe an
abelian subgroup of Gsuch that C∩Ais nonempty, and let Kbe the fixed field of A.
Let g∈C∩A, and let CA=CA(g)denote the conjugacy class of Awhich contains g.
If x≥2, then
πC(x, L/F )−|C|
|G||A|
|CA|πCA(x, L/K)≤|C|
|G|nLx1/2+2
log 2 log DL.
Proof of Theorem 1.1.Let L/F be a Galois extension of number fields and let Cbe a
conjugacy class of the Galois group G= Gal(L/F ). Let Abe an abelian subgroup of G
such that C∩Ais nonempty, and let Kbe the fixed field of A. We apply Lemma 3.1
to the abelian extension L/K. If (3.1) holds, then
nLx1/2=nK[L:K]x1/2≤x
200 log x.(3.3)
By the conductor discriminant formula we have that DL≤DKQ[L:K]. Thus if (3.1)
holds, then
2
log 2 log DL≤2
log 2(log DK+ [L:K] log Q)≤x
200 log x.(3.4)
Thus by Lemmas 3.1 and 3.2 we have that if (3.1) holds, then
πC(x, L/F )≤11.3|C|
|G|
x
log x(3.5)
completing the proof of Theorem 1.1.□
To prove Lemma 3.1, we utilize a version of the Selberg sieve developed by Weiss
[Wei83]. First, we need explicit bounds on certain sums over ideals, which we develop
in Section 4. We use these bounds in Section 5, where we implement the Selberg sieve.
4. Sums over integral ideals
Let aH be a coset of I(m)/H and nan integral ideal coprime to fL/K . Define
A(x;a, n):=X
a∈aH
n|a
ϕlog N(a)
log x,(4.1)
where ϕis the test function described in Lemma 2.2. We obtain estimates for A(x;a, n)
in Section 4.1.

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 15
The second quantity we consider is
V(z):=X
N(a)≤z
1
N(a).(4.2)
We establish lower bounds for V(z) in Section 4.2.
4.1. Bounding A(x;a, n).We begin by computing a smoothed character sum over
integral ideals.
Lemma 4.1. Let x≥3and ϵ, δ ∈(0,1/2). Let χbe a ray class character of Kmodulo
m, and let nbe an integral ideal coprime to m. Let Zm(δ)be defined as in (2.17). If
E(x) := Zm(δ)8
3π
1 + δ
δ(1 −δ)(1 + δ−1)nKeδϵDKQ
(2π)nK1/22nK/2h2
ϵ(1 + ϵ)inK/2+1nnK/2
Kxδ,
(4.3)
then
X
a
n|a
χ(a)ϕlog Na
log x−δχ0(χ)χ(n)φK(m)
Nm
κK
Nn·log x·F(−log x)≤E(x),
Proof. By Laplace inversion, for δ > 0 we have
X
a
n|a
χ(a)ϕlog Na
log x=χ(n)log x
2πi Z1+δ+i∞
1+δ−i∞
L(s, χ)
(Nn)sF(−slog x)ds.
We shift the line of integration from Re(s) = 1+δto Re(s) = δ, justifying the vanishing
of the horizontal integrals by rapid decay of the function F(−slog x) as |t|→∞. In
doing so, we pick up a residue at s= 1, so the above is equal to
δχ0(χ)χ(n)φK(m)
Nm
κK
Nnlog x·F(−log x) + χ(n)log x
2πi Zδ+i∞
δ−i∞
L(s, χ)
(Nn)sF(−slog x)ds.
To estimate the integral, we first express L(s, χ) in terms of an L-function of a
primitive character. If eχis the primitive character which induces χ, we have that
L(s, χ) = L(s, eχ)Y
p|m
(1 −eχ(p)Np−s).(4.4)
We can bound the product over primes in (4.4) by Zm(σ), defined in (2.17). Thus we
have
χ(n)log x
2πi Zδ+i∞
δ−i∞
L(s, χ)
(Nn)sF(−slog x)ds≤log x
2πZm(δ)Zδ+i∞
δ−i∞|L(s, eχ)||F(−slog x)|ds.
(4.5)
We split the remaining integral into two pieces, the first with |t| ≤ Mand the second
with |t|> M, for some constant M > 0 which will be specified later. We bound the

16 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
contribution from L(s, χ) with Lemma 2.1 and the contribution from F(−slog x) with
Lemma 2.2(d). Using these bounds and (4.5) gives
χ(n)log x
2πi Zδ+i∞
δ−i∞
L(s, χ)F(−slog x)ds
≤Zm(δ)1
πDχ
(2π)nK1/21 + δ
1−δ(1 + δ−1)nKeδϵxδ(1 + x−δ/2)
×hZM
0
|δ+1+it|nK/2
|δ+it|dt +2ℓ
ϵnK/2+1 Z∞
M
|δ+1+it|nK/2
|δ+it|nK/2+2 dti.
We bound the first integral by
ZM
0
|δ+1+it|nK/2
|δ+it|dt ≤1
δZM
0
(δ+1+t)nK/2dt
≤1
δ(nK/2 + 1)(δ+1+M)nK/2+1.
We set ℓ=⌈nK/2+1⌉and substitute u= 1 + t−1to bound the second integral by
Z∞
M
|δ+1+it|nK/2
|δ+it|α+1 dt ≤Z∞
M1 + 1
δ+it
nK/2|δ+it|−2dt
≤Z∞
M1 + 1
tnK/2t−2dt
≤Z1+M−1
1
unK/2du
≤1
nK/2+1h(1 + M−1)nK/2+1 −1i.(4.6)
Our bound on the sum of the two integrals is then
1
nK/2+1h1
δ(δ+1+M)nK/2+1 +2ℓ
ϵnK/2+1(1 + M−1)nK/2+1 −1i.
If we choose Mso that
δ+1+M < 2ℓ
ϵ,(4.7)
we can bound the sum of the two integrals (using δ < 1) by
1
(nK/2 + 1)δh2ℓ
ϵ(1 + M−1)inK/2+1.(4.8)
Choosing M=1
ϵsatisfies (4.7). Then, using nK≥1 and recalling that ℓ=⌈nK/2⌉+1 ≤
(nK+ 3)/2≤2nK, we can bound (4.8) by
1
δ·4
3(2nK)nK/2h2
ϵ(1 + ϵ)inK/2+1.
The error term is then bounded by

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 17
Zm(δ)1
πdχ
(2π)nK1/21 + δ
1−δ(1 + δ−1)nKeδϵxδ(1 + x−δ/2)1
δ·4
3(2nK)nK/2h2
ϵ(1 + ϵ)inK/2+1
as desired. □
We now sum over integral ideals in a given coset of the ray class group via Lemma 4.1.
Lemma 4.2. Let m=fL/K , let H= ker(FL/K), and let aH be a coset in I(m)/H. Let
nbe an integral ideal coprime to m. Then
X
a∈aH
n|a
ϕlog N(a)
log x−1
[L:K]
φK(m)
N(m)
κK
N(n)·log x·F(−log x)≤E(x),
where E(x)is as in (4.3).
Proof. By character orthogonality applied to the group I(m)/H, we have that
X
a∈aH
n|a
ϕlog N(a)
log x=1
[L:K]X
χ(H)=1
χ(aH)X
a
n|a
χ(a)ϕlog N(a)
log x.
Note that χ(a) = 0 is defined if aand mare not coprime. Summing over each character
using the previous lemma gives the desired result. □
4.2. Bounding V(z).We proceed similarly to the proof of Lemma 4.1. In particular,
we bound V(z) by a sum of test functions, and apply an inverse Laplace transform.
Our main result is the following.
Lemma 4.3. Let 0< ω < 1/2and let η, ϵ ∈(0,1/2). Set z=xω. Set
ec11 =21/2
e1/2πη+ 1
η(1 −η)eϵω/2h1
ϵω (1 + ϵω)(1 + eϵ)i3/2
ec12 =e1/2
π1/2η+ 1
ηh1
ϵω (1 + ϵω)(1 + eϵ)i1/2.(4.9)
Denote by κKthe residue of ζK(s)at s= 1. If
x≥eϵec11 ec12nKnKnK/2D1/2
K2
(1−η)ω,(4.10)
then
V(z)≥κKω
2log x. (4.11)
Proof. Choose z′such that
log z
log z′≥1 + ϵ
log x
This arises from the support condition supp ϕ⊂[1
2−ϵ
log x,1 + ϵ
log x]. Rearranging gives
log z′≤log z
1 + ϵ
log x
.

18 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
Since (1 + ϵ/ log x)−1>1−ϵ/ log x, we can set
z′=z1−ϵ/ log x=ze−ϵω = (xe−ϵ)ω.(4.12)
We have the inequality
V(z) := X
N(a)≤z
1
N(a)≥1 + X
a
1
N(a)ϕlog N(a)
log z′.
By Laplace inversion, since 0 < η ≤1/2, we have that
X
a
1
N(a)ϕlog N(a)
log z′=log z′
2πi Zη+i∞
η−i∞
ζK(s+ 1)F(−slog z′)ds
=κKF(0) ·log z′+log z′
2πi Z−1+η+i∞
−1+η−i∞
ζK(s+ 1)F(−slog z′)ds.
Using the definition of z′and Lemma 2.2(e), we can bound the contribution of the
residue from below as
κKF(0) ·log z′=κKω(log x−ϵ)1
2+ϵ
log x
≥κKω
2log x.
As before, we break up the integral into two parts, the first with |t| ≤ Mand the
second with |t|> M. In both regions we use Lemma 2.1 to bound ζK(s+ 1). We use
Lemma 2.2(g) to bound the integrand with |t| ≤ M, and Lemma 2.2(f) for |t|> M.
Applying these bounds gives
Z−1+η+i∞
−1+η−i∞
ζK(s+ 1)F(−slog z′)ds(4.13)
≤1 + η
1−η(1 + η−1)nKDK
(2π)nK1/22eϵω/2(z′)(−1+η)/2
log z′
×hZM
0
|η+1+it|nK/2
|η+it|2
2− |η+it|ϵω/ℓ ℓdt +ℓ(1 + eϵ)
ϵℓZ∞
M
|η+1+it|nK/2
|η+it|ℓ+1 dti.
(4.14)
To bound these integrals, we set ℓ=⌈nK/2⌉+ 1, so that nK/2≤ℓ−1. We bound
the first integral by
ZM
0
|η+1+it|nK/2
|η+it|2
2− |η+it|ϵω/ℓ ℓdt ≤1
η2
2−(η+M)ϵω/ℓ ℓZM
0
(η+1+t)ℓ−1dt
≤1
η2
2−(η+M)ϵω/ℓ ℓ1
ℓ(η+1+M)ℓ.

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 19
We bound the second integral using the fact that |η+ 1 + it|nK/2≤ |η+ 1 + it|ℓ−1and
then integrating as in (4.6)
Z∞
M
|η+1+it|nK/2
|η+it|l+1 dt ≤Z∞
M
|η+1+it|ℓ−1
|η+it|l+1 dt
≤Z∞
M1 + 1
η+it
ℓ−1|η+it|−2dt
≤1
ℓh(1 + M−1)ℓ−1i.
So overall, our bound on the sum of the two integrals in (4.14) is
1
ℓh1
η2(η+1+M)
2−(η+M)ϵω/ℓ ℓ+ℓ(1 + eϵ)
ϵω ℓ(1 + M−1)ℓ−1i
We want to choose Msmall enough so that
2(η+1+M)
2−(η+M)ϵω/ℓ <ℓ(1 + eϵ)
ϵω (4.15)
and we can then bound the entire quantity (using η < 1) by
1
ℓη hℓ
ϵω (1 + M−1)(1 + eϵ)iℓ.(4.16)
Setting M=1
ϵω satisfies (4.15). Then, by recalling that ℓ=⌈nK/2⌉+ 1 ≤2nKand
using that nK≤enK−1, we can bound (4.16) by
1
ηnnK/2
K2nK/221/2enK/2e−1/2h1
ϵω (1 + ϵω)(1 + eϵ)iℓ
So in all, our bound for the integral in (4.13) is
(z′)(−1+η)/2ec11 ec12nKnnK/2
KD1/2
K
where
ec11 =21/2
e1/2π
η+ 1
η(1 −η)eϵω/2h1
ϵω (1 + ϵω)(1 + eϵ)i3/2
ec12 = (1 + η−1)(2π)−1/221/2e1/2h1
ϵω (1 + ϵω)(1 + eϵ)i1/2.
We need the error term to be less than 1, which is satisfied when
z′≥ec11 ec12nKnnK/2
KD1/2
K2/(1−η).
Recalling that z′= (xe−ϵ)ωcompletes the proof. □

20 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
5. Implementing the Selberg sieve
Let L/K be an abelian extension of number fields. In this section we prove an
upper bound for πC(x, L/K ) via the Selberg sieve. Our implementation closely follows
the classical method of proof of the Brun–Titchmarsh theorem via the Selberg sieve
(see [CM+06] for one such proof). We apply the analogue for number fields outlined
in [Wei83]. Our main result is the following.
Lemma 5.1. Let L/K be an abelian extension of number fields and Ca conjugacy
class (which will necessarily be a single element) of Gal(L/K). Let πC(x, L/K)be as
in (1.3),E(x)be as in (4.3), and 0< ω, η, ϵ, γ < 1/2. If xsatisfies the range condition
(4.10), then
πC(x, L/K)≤2.52nK√x
log x+2eϵx
ω[L:K] log x+E(x)(1 + γ−1)nKxω(1+γ)2,(5.1)
Proof. Let aH :=F−1
L/K (C) and recall (2.2). We would then like to bound
πC(x, L/K) = πaH (x, L/K) = X
punramified
p∈aH
NK/Q(p)≤x
1.
To begin, we split the sum into two pieces as follows.
πaH (x, L/K)≤X
p∈aH
N(p)≤x
1 = X
p∈aH
N(p)≤√x
1 + X
p∈aH
√x<N(p)≤x
1.(5.2)
Since at most nKprime ideals pof Klie over a given rational prime p, we can use the
universal bound Pp≤x1≤1.26 x
log xfrom Corollary 1 of [RS62] to bound the first sum
in (5.2) by X
p∈aH
N(p)≤√x
1≤X
p≤√xX
p|⟨p⟩
1≤nKX
p≤√x
1≤2.52nK√x
log x.(5.3)
We bound the second term via the Selberg sieve. For an integral ideal a, we define
the quantity P−(a) to be the least norm of a prime ideal factor of a. Let 0 < z < √x.
Let Szbe the set of those integral ideals satisfying P−(a)> z; that is, integral ideals
with no prime ideal factor having norm ≤z. It follows that
X
p∈aH
√x<N(p)≤x
1≤X
a∈aH∩Sz
√x<N(a)≤x
1.
For the test function ϕdefined in Lemma 2.2, we have that ϕ(t) = 1 for t∈[1
2,1] and
ϕis nonnegative, so X
a∈aH∩Sz
√x<N(p)≤x
1≤X
a∈aH∩Sz
ϕlog N(a)
log x.
Now, consider any real-valued function λon integral ideals of OKsatisfying
(a) λ(OK) = 1;

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 21
(b) λ(b) = 0 unless bis squarefree, coprime to mand N(b)≤z;
(c) |λ(b)| ≤ 1 for all b.
We will eventually specify the choice of such a function.
Denote by [b1,b2] and (b1,b2) the least common multiple and greatest common divisor
of two integral ideals b1,b2. Also denote by µKthe M¨obius function on integral ideals
associated to the number field K, which is defined for an integral ideal aof OKas
µK(a) := ((−1)ω(a)ais squarefree,
0ais not squarefree.
Note that if a∈Sz, then b|aimplies b=OKor N(b)> z, which by conditions (a)
and (b) implies that b=OKor λ(b) = 0. Hence, we have
X
a∈aH∩Sz
ϕlog N(a)
log x≤X
a∈aH
ϕlog N(a)
log xX
b|a
λ(b)2
=X
b1,b2
λ(b1)λ(b2)X
a∈aH
[b1,b2]|a
ϕlog N(a)
log x.(5.4)
Now by Lemma 4.2, we have for n,mcoprime that
X
a∈aH
n|a
ϕlog N(a)
log x≤1
[L:K]
φK(m)
N(m)
κK
N(n)(log x)·F(−log x) + |E(x)|,
If [b1,b2] is not coprime to m, then λ(b1)λ(b2) = 0 by condition (b), so (5.4) is bounded
by
X
b1,b2
λ(b1)λ(b2)1
[L:K]
φK(m)
N(m)
κK
N([b1,b2])(log x)·F(−log x) + |E(x)|.
Since b1b2= (b1,b2)[b1,b2], we can rewrite this as
X
b1,b2
λ(b1)λ(b2)1
[L:K]
φK(m)
N(m)
κKN((b1,b2))
N(b1)N(b2)(log x)·F(−log x) + E(x).
Applying the identity N(a) = Pb|aφK(b) and |λ(b)| ≤ 1 yields
κK(log x)F(−log x)
[L:K]
φK(m)
N(m)X
b1,b2
λ(b1)λ(b2)
N(b1)N(b2)X
a|b1,a|b2
φK(a) + X
b1,b2λ(b1)λ(b2)E(x)
≤κK(log x)F(−log x)
[L:K]
φK(m)
N(m)X
a
φK(a)X
a|b
λ(b)
N(b)2+E(x)X
b
N(b)≤z
12.(5.5)
By considering the M¨obius inversion pair
ξ(a) = X
b|a
λ(b)
N(b),λ(a)
N(a)=X
b
µK(b)ξ(ab),

22 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
we seek to choose λ(equivalently, to choose ξ) such that the quadratic form PaφK(a)ξ(a)2
is minimized. Using the same choice as in [Wei83, p. 73] and recalling the definition
of V(z) in (4.2) yields the following bound from [Wei83, Lemma 3.6, eq. (iv)], which
states that φK(m)
N(m)X
a
φK(a)X
a|b
λ(b)
N(b)2≤1
V(z).
Substituting this into equation (5.5) yields the bound
κK(log x)F(−log x)
[L:K]V(z)+E(x)X
N(b)≤z
12.
By Lemma 2.2(e) the above is bounded by
κKeϵx
[L:K]V(z)+E(x)X
N(b)≤z
12.
Now, by the lower bound for V(z) from Lemma 4.3 (in the appropriate range for x
specified in that lemma, with z=xω), the above is bounded by
2eϵx
ω[L:K] log x+E(x)X
N(b)≤z
12.
Applying Lemma 2.3 and substituting z=xωyields a bound (for 0 < γ ≤1) of
X
p∈aH
√x<N(p)≤x
1≤2eϵx
ω[L:K] log x+E(x)(1 + γ−1)nKxω(1+γ)2.(5.6)
Combining (5.3) and (5.6) into (5.2) gives the desired result. □
6. Brun–Titchmarsh for abelian extensions
In this section we prove Lemma 3.1. We begin by bounding the error terms in Lemma
5.1. We bound the first term in (5.1) as
2.52nK√x
log x≤x
100[L:K] log x(6.1)
in the range
x≥(252nK[L:K])2.(6.2)
Next we consider the last term in (5.1). Let E(x) be as in (4.3). Then we define
C(δ, ϵ, γ) = E(x)(1 + γ−1)2nK[L:K]x−δ,(6.3)
which does not depend on x. We want to choose xlarge enough so that
E(x)(1 + γ−1)2nKx2ω(1+γ)≤x
[L:K] log x,
which is equivalent to
x1−δ−2ω(1+γ)
log x≥C(δ, ϵ, γ).(6.4)

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 23
By Lemma 2.4, we find that if
x≥C(δ, ϵ, γ)
1−δ−2ω(1 + γ)ν,(6.5)
where
ν=1
1−δ−2ω(1 + γ)1 + 2
log(C(δ, ϵ, γ)/(1 −δ−2ω(1 + γ))) 1/2,(6.6)
then (6.4) is satisfied. Combining the bounds in this section with Lemma 4.3 yields our
first version of the Brun-Titchmarsh theorem for abelian extensions.
Lemma 6.1. Let L/K be an abelian extension of number fields, let Cbe a conjugacy
class (which will necessarily be a single element) of Gal(L/K), and let πC(L/K )be
the prime–counting function given in (1.3). Let 0< ϵ, δ, γ, η, ω < 1/2be such that
1−δ−2ω(1 + γ)>0, and let c11, c12 be as in (4.9),C(δ, ϵ, γ)be as in (6.3), and νbe
as in (6.6). If
x≥max eϵec11 ec12nKnKnK/2D1/2
K2
(1−η)ω,(252nK[L:K])2,C(δ, ϵ, γ)
1−δ−2ω(1 + γ)ν
(6.7)
then
πC(x, L/K)≤1.01 + 2eϵ
ωx
[L:K] log x.(6.8)
We state the above version of the Brun-Titchmarsh theorem as it allows us flexibil-
ity in choosing parameters when the field extension L/K is known, as in Section 7.
For L/K a general abelian extension of number fields, we derive an explicit form of
Lemma 6.1 by fixing parameters. Note that different choices of parameters may im-
prove the dependence of the range on certain field constants (such as nK, DK,Q) at the
expense of others.
Proof of Lemma 3.1.Let δ= 1/10, η= 1/21, ϵ=ω= 1/4, and γ= 1/5. We wish
to obtain a range for xin terms of nK, DK, and Qvia Lemma 6.1. To do so, we
use Lemma 2.5 with the parameter ϵ= 1 to bound [L:K] and Lemma 2.7 with the
parameters M= 400 and b= (2 log(1 + M−δ))−1to bound Zm(δ). Doing so gives
C(δ, ϵ, γ)≤e7.36e22.85n(DKQ)2nn/2.(6.9)
Applying these bounds to (6.6) gives ν≤4.189. Now, combining the bounds given in
Lemma 6.1 and substituting our parameters gives the desired result. □
7. Upper bounds for the Lang–Trotter conjecture
In this section we apply our Brun-Titchmarsh results to study coefficients of modular
forms. Let f(z)∈ Snew
kf(Γ0(Nf)) be a non-CM cusp form of even weight kf≥2, level
Nf, trivial nebentypus, and integral coefficients af(n) which is an eigenform for all
Hecke operators and Atkin–Lehner operators. Let
πf(x, a):= #{p≤xprime, p ∤Nf,and af(p) = a}.

24 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
We use a sieving procedure to bound πf(x, a) by a sum of auxiliary prime-counting
functions πf(x, a, ℓ) (for a suitably chosen collection of primes ℓ) which count coefficients
af(p) congruent to a(mod ℓ), with some additional conditions. The projection onto
Fℓof the ℓ-adic Galois representation ρf,ℓ attached to fallows us to express these
conditions as a condition on the image of Frobpunder the residual representation. This
allows the count of rational primes pwith given af(p) to be compared with the count of
prime ideals with prescribed Frobenius class in a certain abelian extension of number
fields. This extension is a subextension of the fixed field of ker ρf,ℓ in Q, through which
ρf,ℓ factors. This enables us to compute an explicit upper bound for πf(x, a) using
Lemma 6.1.
7.1. Sieving πf(x, a)by primes. For a prime number p, define
ωp= (af(p)2−4pkf−1)1/2.(7.1)
We know from Deligne’s proof of the Weil conjectures that |af(p)| ≤ 2p(kf−1)/2for all
p, so Q(ωp) is an imaginary quadratic extension of Q. For an odd prime ℓ, set
πf(x, a;ℓ)=#np≤xprime, p ∤Nf, af(p)≡a(mod ℓ),a2−4pkf−1
ℓ= +1o.
Here, ( ·
ℓ) is the Legendre symbol; in other words the latter condition says ℓsplits in
Q(ωp). Our sieve result is the following lemma adapted from [Wan90, Lemma 4.1].
Lemma 7.1. Let x, t > 0be integers, and let ℓ1< ℓ2<· ·· < ℓtbe todd primes, each
less than x. Assume furthermore that gcd(ℓj−1, kf−1) = 1 for all j= 1, . . . , t. Then
πf(x, a)≤
ℓ
X
j=1
πf(x, a;ℓj) + √2·x
2t/2+√2·2t/2ℓt
t+ 2t+70
2t.(7.2)
Proof. Let ℓ1< ℓ2<·· · < ℓtbe as above, and let Mt(x) be the number of primes p≤x
such that a2−4pkf−1
ℓj= +1 for all j= 1, . . . , t, that is, the number of primes p≤x
that are not counted in any of the πf(x, a;ℓj). Then
πf(x, a)≤
t
X
j=1
πf(x, a, ℓj) + Mt(x).(7.3)
To estimate Mt(x), we introduce a weight function wt(p) on the set of primes. Given
a prime p, suppose that a2−4pkf−1
ℓ= +1,0,−1 for, respectively, t1(p), t2(p), t3(p) of
the primes ℓj. So t1+t2+t3=t. Define
wt(p) = 1
2t
t
Y
j=1 1−a2−4pkf−1
ℓj=(0 if t1(p)= 0,
2−t2(p)if t1(p) = 0,
and let
Wt(x) = X
p≤x
wt(p).

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 25
Then
Wt(x)≤1
2tX
n≤x
t
Y
j=1 1−a2−4nkf−1
ℓj
=X
n≤x1
2t+ max
d≥2
d|LtX
n≤xa2−4nkf−1
d,
where Lj=ℓ1··· ℓjfor j= 1, . . . , t, so Lt≤ℓt
t. The last sum is a character sum mod d
since we assume that gcd(ℓj−1, kf−1) = 1 for all j. We trivially bound this character
sum above by d≤Lt. Thus,
Wt(x)≤x
2t+ℓt
t.
Let M′
t(x) be the number of primes p≤xsuch that t2(p)≥t/2 (that is, a2−4pkf−1
ℓj= 0
So
Mt(x)−M′
t(x)≤2⌈t/2⌉Wt(x)≤√2·2t/2Wt(x).
Arguing via the Chinese remainder theorem, appealing again to our assumption that
gcd(ℓi−1, kf−1) = 1, we have that
M′
t(x)≤t
⌈t/2⌉x
ℓ1··· ℓ⌈t/2⌉
+ 1≤2tx
ℓ1··· ℓ⌈t/2⌉
+ 2t.
Therefore
Mt(x)≤√2·x
2t/2+√2·2t/2ℓt
t+ 2tx
L⌈t/2⌉
+ 2t.(7.4)
Since ℓj≥24except possibly for the first 5 odd primes, we have that
L⌈t/2⌉≥3·5·7·11 ·13
165·⌈t/2⌉
Y
i=1
16 ≥70−1·4t,
and inserting (7.4) into (7.3) gives the lemma. □
For our application, we state the following special case of Lemma 7.1.
Corollary 7.2. Let r, x > 1, let t=⌈(2r/ log 2) log log x⌉, and let ℓ1< ℓ2<· ·· < ℓtbe
todd primes, each less than exp( log x
2t). Assume furthermore that gcd(ℓj−1, kf−1) = 1
for all j= 1, . . . , t. Then
πf(x, a)≤2r
log 2 log log x+ 1max
1≤t≤jπf(x, a;ℓj)
+x√2
(log x)r+2x1/2
(log x)r+ 2(log x)2r+ 35.(7.5)
Proof. Observe that we have
2r
log 2 log log x≤t < 2r
log 2 log log x+ 1
with this choice of t, and use this to bound all terms in (7.2) from above. □

26 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
7.2. Reduction to a Chebotarev problem. Let ℓbe any odd prime, and let Fℓbe
the field of ℓelements. For any prime p, let Frobpbe the Frobenius automorphism
of Gal(Q/Q) at p. By a theorem of Deligne, one associates to each newform f∈
Snew
kf(Γ0(Nf)) an ℓ-adic representation, and, by projection onto Fℓ, a representation
ρf,ℓ : Gal(Q/Q)→GL2(Fℓ),
which is unramified outside Nfℓ, and such that for all primes p∤Nfℓ, we have that
tr(ρf,ℓ(Frobp)) ≡af(p) (mod ℓ) and det(ρf,ℓ(Frobp)) ≡4pkf−1(mod ℓ).
Let L=Lℓbe the subfield of Qfixed by ker ρf,ℓ. It is known that there exists ℓ0
depending on fsuch that, if ℓ > ℓ0, then L/Qis a Galois extension, unramified outside
of Nfℓ, whose Galois group is G={g∈GL2(Fℓ)|det(g)∈(F×
ℓ)kf−1}. Hence, if
gcd(ℓ−1, kf−1) = 1, then G= GL2(Fℓ); that is, the representation ρf,ℓ is surjective.
From now on we let ℓ > ℓ0be a prime number with gcd(ℓ−1, kf−1) = 1, and let
C={A∈G|tr(A)≡a(mod ℓ) and tr(A)2−4 det(A)∈Fℓis a nonzero square}.
The latter condition is equivalent to saying that the matrix Ahas distinct eigenvalues
in Fℓ. The set Cis stable under conjugation in G, and can be expressed as
C=[
γ∈Γ
CG(γ)
where
Γ = nα0
0β∈Gα=β, α +β≡a(mod ℓ)o,
and CG(γ) denotes the conjugacy class in Gthat contains γ. Let Bdenote the Borel
subgroup of upper triangular matrices in G, and let LBbe the subfield of Lfixed by
B. Let Hbe the subgroup of Bconsisting of matrices whose eigenvalues are both
equal, and let LHbe the subfield of Lfixed by H. Let Ube the subgroup of unipotent
elements in B, and let LUbe the subfield of Lfixed by U. We have
{1} ⊂ U⊂H⊂B⊂G= GL2(Fℓ).
Note that Uand Hare normal subgroups of Band that B/U ,B/H are abelian. We
have the tower of field extensions
Q⊂LB⊂LH⊂LU⊂L,
with LU/LBand LH/LBboth abelian extensions. Finally, let C′be the image of C∩B
in B/U and let C′′ be the image of C∩Bin B/H .
Before stating the reduction to counting prime ideals in conjugacy classes of Galois
groups, we collect some basic facts about the cardinality of these groups and conjugacy
classes.
Proposition 7.3. We have |C′| ≤ ℓ,
|C′|
|B/U |≤1
ℓ−1and |C′′|
|B/H |=1
ℓ−1,
and [LB:Q] = ℓ+ 1,[LH:LB] = ℓ−1, and [LU:LH] = ℓ.

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 27
Proof. By [Zyw15, Proof of Lemma 4.4], we have |G|= (ℓ−1)2(ℓ+ 1)ℓ,|B|= (ℓ−1)2ℓ,
|H|= (ℓ−1)ℓ,|U|=ℓ−1, |C′| ≤ ℓ, and |C′′|= 1. □
We now compare πf(x, a;ℓ) to a prime-counting function of the form πC(x, L/K).
Namely, for all a∈Z, we will bound πf(x, a;ℓ) in terms of πC′(x, LU/LB). However,
when a≡0 (mod ℓ), the conjugacy class Cconsists of matrices of zero trace over Fℓ,
which allows us to work with the subextension LH/LBand its corresponding conjugacy
class C′′. In view of this we state two versions of the reduction step, one applicable to
all a∈Zand another specialized to the case a≡0 (mod ℓ).
For a given modular form, we may also know additional congruence relations which
its Fourier coefficients must satisfy. For instance, say we know that if af(p) = a, then p
lies in some fixed set Sof residue classes modulo q≥1. This condition imposes further
conditions on the Artin symbol; via our reduction, it translates to counting primes in
a compositum with a q-cyclotomic extension (that is, LU(ζq)/LBor LH(ζq)/LB).
For L/K a Galois extension of number fields with Galois group G, and C⊆Ga
conjugacy class, we set
eπC(x, L/K ):=X
m≥1
NK/Q(p)m≤x
Frobm
p=C
1
m,
a modified version of πC(x, L/K). Then it is clear that πC(x, L/K)≤eπC(x, L/K).
See [Zyw15, Section 2.3] for elegant functorial properties enjoyed by the function eπC.
Lemma 7.4. Let fand ℓbe as above, let Q⊂LB⊂LH⊂LU⊂Lbe constructed
as above, and let abe an integer. Let q≥1be an integer with gcd(q, Nfℓ) = 1, let
S⊆(Z/qZ)×be a set of residue classes modulo q, and suppose we have the following:
for a prime number p, if af(p) = a, then p∈S.
(a) We have
πf(x, a;ℓ)≤eπC′×S(x, LU(ζq)/LB)+1.(7.6)
(b) If a≡0 (mod ℓ), we have
πf(x, 0; ℓ)≤eπC′′ ×S(x, LH(ζq)/LB)+1.(7.7)
Note that if q= 1, then πf(x, a;ℓ)≤eπC′(x, LU/LB)and πf(x, 0; ℓ)≤eπC′′(x, LH/LB).
Proof. From the condition gcd(q, Nfℓ) = 1, the fields Land Q(ζq) are linearly disjoint,
so Gal(L(ζq)/Q) = Gal(L/Q)×Gal(Q(ζq)/Q) = G×(Z/qZ)×. We claim that
πf(x, a;ℓ)≤πC×S(x, L(ζq)/Q)+1.
Indeed, let p≤xbe a prime such that p∤Nfℓ,af(p) = aand a2−4pkf−1is a
square in F×
ℓ. The representation ρf,ℓ is unramified at pand we have tr(ρf,ℓ(Frobp)) ≡
af(p) = a(mod ℓ) and det(ρf,ℓ (Frobp)) ≡pkf−1(mod ℓ). Thus tr(ρf,ℓ (Frobp))2−
4 det(ρf,ℓ(Frobp)) ∈Fℓis a square. We have thus shown that ρf,ℓ(Frobp)⊆C. By
hypothesis, we also know that if af(p) = a, then p∈S(mod q). Adding an extra 1 to
account for the excluded prime p=ℓ, this implies that
πf(x, a;ℓ)≤πC×S(x, L(ζq)/Q)+1
≤eπC×S(x, L(ζq)/Q)+1.

28 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
By [Zyw15, Lemma 2.6(i)], and since LBis the fixed field in L(ζq) of the subgroup
B×(Z/qZ)×, we have that
πf(x, a;ℓ)≤eπ(C∩B)×S(x, L(ζq)/LB)+1.
By [Zyw15, Lemma 2.6(ii)], and since U× {1}is a normal subgroup of B×(Z/qZ)×
with (U× {1})·((C∩B)×(Z/qZ)×)⊆((C∩B)×(Z/qZ)×), we also have that
πf(x, a;ℓ)≤eπC′×S(x, LU(ζq)/LB)+1,
which is part (a). If a≡0 (mod ℓ), then Cconsists of trace zero matrices, and so we
repeat the last line with the subgroup H×{1}in the place of the subgroup U×{1}.□
Now, for a Galois extension of number fields L/K, let P(L/K) denote the set of
rational primes pthat are divisible by some prime ideal pof Kthat ramifies in L.
Define
M(L/K):= 2[L:K]D1/[K:Q]
K·Y
p∈P(L/K )
p. (7.8)
The next lemma transitions from the modified eπCto πC, with explicit remainder term.
Lemma 7.5. Let L/K be a normal extension of number fields with Galois group Gand
let C⊆Gbe a union of conjugacy classes. If x≥4, then
eπC(x, L/K )≤πC(x, L/K)+3.15546nK
x1/2
log x+nKlog M(L/K).
Proof. The lemma follows from inspection of the proof of [Zyw15, Lemma 2.7], using
the identity P∞
m=2 m−2=1
6π2−1 and the universal bound π(x)<1.25506 x
log x.□
7.3. Bounding M(L/K).The combination of Corollary 7.2, Lemma 7.4, and Lemma 7.5
allows us to express πf(x, a) in terms of πC′×S(x, LU(ζq)/LB) or πC′×S(x, LH(ζq)/LB) if
a= 0. Before proceeding, we want to establish bounds for the size of the field constants
M(LU(ζq)/LB) and M(LH(ζq)/LB) in terms of the quantities q,Nf, and ℓ. First we
need the following result due to [Ser81].
Lemma 7.6. We have
D1/[LB:Q]
LB≤rad(Nf)ℓ·(ℓ+ 1)ω(Nf)+1.(7.9)
Proof. An application of Proposition 6 of [Ser81] gives
log DLB≤ℓX
p∈P(LB/Q)
log p+|P(LB/Q)|(ℓ+ 1) log(ℓ+ 1).(7.10)
Since p∈ P(LB/Q) implies p|Nfℓ, we may bound the above sum by log(rad(Nf)ℓ).
Similarly, we may bound |P(LB/Q)|by ω(Nf) + 1, proving the lemma. □
We use this result to prove the next lemma.
Lemma 7.7. Suppose that ℓ>ℓ0with gcd(ℓ−1, kf−1) = 1, and Q⊂LB⊂LH⊂
LU⊂Lare constructed as above, and q≥1is an integer with gcd(q, Nfℓ)=1. Let

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 29
ω(n)denote the number of distinct prime factors dividing a positive integer n, and let
rad(n)denote their product, with rad(1) = 1. We have that
M(LU(ζq)/LB)≤2(ℓ+ 1)ω(Nf)+4 rad(Nf)2φ(q) rad(q) (7.11)
and
M(LH(ζq)/LB)≤2(ℓ+ 1)ω(Nf)+3 rad(Nf)2φ(q) rad(q).(7.12)
Proof. Using the previous lemma with [LU(ζq) : LB] = φ(q)(ℓ−1)ℓgives
M(LU(ζq)/LB) = 2[LU(ζq) : LB]D1/[LB:Q]
LB·Y
p∈P(LU(ζq)/LB)
p
≤2φ(q)(ℓ−1)ℓ·rad(Nf)ℓ·(ℓ+ 1)ω(Nf)+1 Y
p∈P(LU(ζq)/LB)
p.
Since p∈ P(LU(ζq)/LB) implies p|qNfℓ, where q,ℓand Nfare assumed pairwise
coprime, we may bound the product by rad(qNf)ℓ.
The proof for M(LH(ζq)/LB) is practically identical; the only change is due to the
fact that [LH(ζq) : LB] = φ(q)(ℓ−1). □
Observe that the bound (7.12) that we achieve for M(LH(ζq)/LB) appears with one
less power of ℓ+ 1 than (7.11). Because of this, working with πC′′(x, LH(ζq)/LB) will
be advantageous when possible.
Finally, if L/K is an abelian extension, we can relate M(L/K) to the quantity
DKQ(L/K).
Lemma 7.8. Let L/K be an abelian extension of number fields. Then
DKQ(L/K)≤M(L/K)
22[K:Q].
Proof. By [MMS88, Proposition 2.5], we have
Q(L/K)≤[L:K]Y
p∈P(L/K )
p2[K:Q].
We multiply both sides by D2
Kand use the definition (7.8) of M(L/K). □
7.4. Lang–Trotter type bounds. Let f(z) be a non-CM cusp form of even weight
kf≥2, level Nf, and trivial nebentypus with integer coefficients af(n) which is an
eigenform for all Hecke operators and Atkin–Lehner operators. Suppose furthermore
that for some integer q≥1, gcd(q, ℓ) = 1 and subset S⊆(Z/qZ)×, it is known that
if af(p) = a, then p∈S(mod q). Finally, suppose that the Galois representation ρf,ℓ
is surjective for all primes ℓ > ℓ0with gcd(ℓ−1, kf−1) = 1. For such ℓ, constructing
LB⊂LH⊂LUas before, we have the bounds for M(LU(ζq)/LB) and M(LH(ζq)/LB)
provided in Lemma 7.7.
In this subsection, we are interested in estimating the prime counting function πf(x, a).
Of particular interest is πf(x, 0), as this quantity is related to the non-vanishing of the
Fourier coefficients af(n) of f(see Proposition 8.4 below). First, using Lemma 6.1, we
develop an estimate for πC′′×S(x, LH(ζq)/LB) and πC′×S(x, LU(ζq)/LB) in terms of only
ℓand q.

30 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
Lemma 7.9. Let ℓ > max{ℓ0,5}be a prime with gcd(ℓ−1, kf−1) = 1 and qa positive
integer such that gcd(q, ℓ)=1. Let
c13 = max(62 + 4.2 log rad(Nf),4.2(2.9 + log φ(q))),
c14 = max(42,4.2(5.8 + log φ(q) + log rad(q) + log 2(1 + ω(q) + ω(Nf)))),
c15 = 4.2(ω(Nf)+3.5).
If
x≥ec13 ec14(ℓ+1)(ℓ+ 1)c15(ℓ+1) ,(7.13)
then
πC′′×S(x, LH(ζq)/LB)≤11.29 |S|x
φ(q)(ℓ−1) log x.(7.14)
Proof. We wish to express all the quantities present in Lemma 6.1 in terms of ℓand q.
For the extension LH(ζq)/LB, we have that
nLB=ℓ+ 1,
[LH(ζq) : LB] = φ(q)(ℓ−1) ≤φ(q)eℓ+1−3.
By Lemmas 7.8 and 7.7, we have that
DLBQ(LH(ζq)/LB)≤(ℓ+ 1)2(ω(Nf)+3)(ℓ+1)(φ(q) rad(q))2(ℓ+1).(7.15)
By Lemma 7.6, we also have
DLB≤(rad(Nf)(ℓ+ 1))ℓ+1(ℓ+ 1)(ω(Nf)+1)(ℓ+1).(7.16)
Lastly, to bound Zm(δ), note that the Artin conductor for LH(ζq)/LBis divisible only
by primes in LBwhich ramify in LH(ζq). By considering ramification of rational primes
in Land Q(ζq) separately, P(LH(ζq)/LB) is a subset of the primes in LBwhich lie over
primes in P(L(ζq)/Q), i.e. primes in LBwhich divide qNfℓ. So, we obtain the bound
ω(m)≤ω(qNfℓ)[LB:Q] = (1 + ω(q) + ω(Nf))(ℓ+ 1),
which gives
Zm(δ)≤2ω(m)≤2(1+ω(q)+ω(Nf))(ℓ+1).(7.17)
Now, applying these bounds to Lemma 6.1 with the parameters δ= 1/10, η= 1/21,
ϵ=ω= 1/4, and γ= 1/5 gives
eϵec11 ec12nLBnLBnLB/2D1/2
LB2
(1−η)ω≤e62e42(ℓ+1) rad(Nf)4.2(ℓ+ 1)4.2(ℓ+1)(ω(Nf)+3)
(252nLB[LH(ζq) : LB])2≤e12e4(ℓ+1) φ(q)2
C(δ, ϵ, γ)
1−δ−2ω(1 + γ)ν≤
e2.9φ(q)(φ(q) rad(q))ℓ+1e5.8(ℓ+1)2(1+ω(q)+ω(Nf))(ℓ+1)(ℓ+ 1)(ω(Nf)+3.5)(ℓ+1) 4.2.
Taking the maximum of these three expressions gives the desired result. □
A similar result holds for πC′×S(x, LU(ζq)/LB).

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 31
Lemma 7.10. Let ℓ > max{ℓ0,5}be a prime with gcd(ℓ−1, kf−1) = 1 and qa
positive integer such that gcd(q, ℓ) = 1. Let
c16 = max(62 + 4.2 log rad(Nf),4.2(0.9 + log φ(q))),
c17 = max(42,4.2(6.8 + log φ(q) + log rad(q) + log 2(1 + ω(q) + ω(Nf)))),
c18 = 4.2(ω(Nf)+4.5).
If
x≥ec16 ec17(ℓ+1)(ℓ+ 1)c18(ℓ+1) ,(7.18)
then
πC′×S(x, LU(ζq)/LB)≤11.29 |S|x
φ(q)(ℓ−1) log x.(7.19)
Proof. The proof is the same as that of Lemma 7.9, but we instead use the bound
DLBQ(LU(ζq)/LB)≤(ℓ+ 1)2(ω(Nf)+3)(ℓ+1)(φ(q) rad(q))2(ℓ+1) (7.20)
from Lemmas 7.8 and 7.7 and
[LU(ζq) : LB] = φ(q)ℓ(ℓ−1) ≤φ(q)e2(ℓ+1)−5.(7.21)
□
Note that if the range conditions (7.13) or (7.18) hold with ℓ≥5, we have that
x≥e272, regardless of Nfand q. Therefore, this will be established as an assumption
throughout.
To conclude, we derive a general-purpose bound for πf(x, 0) and πf(x, a) for a new-
form fwith the aforementioned specifications. We can do so using Lemmas 7.9 or 7.10
combined with Corollary 7.2, Lemma 7.4, and Lemma 7.5.
Theorem 7.11. Let a∈Z,A > 272 and 1< m < 10 be constants. Suppose x≥eA
and ℓ=ℓ1is a prime number, and set
t=l2r
(log 2) log log xm.(7.22)
Suppose that the following also hold:
(A) ℓ > max{ℓ0,5}and gcd(ℓ−1, kf−1) = 1 and gcd(ℓ, q) = 1,
(B) x≥ec16 ec17(ℓ+1)(ℓ+ 1)c18(ℓ+1) , where c16,c17 ,c18 are as in (7.18). If a= 0,
we may instead take x≥ec13ec14 (ℓ+1)(ℓ+ 1)c15 (ℓ+1), where c13 ,c14,c15 are as in
(7.13).
(C) ℓ < 1
2exp(log x
2t),
(D) There exist tprimes ℓ=ℓ1< ℓ2<·· · < ℓtin the interval [ℓ, 2ℓ)such that
gcd(ℓj−1, kf−1) = 1 and gcd(ℓj, q) = 1.
We have that
πf(x, a)≤34.7r|S|
φ(q)(ℓ−1)
xlog log x
log x+ 1.42 x
(log x)r.(7.23)

32 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
Proof. We apply Lemma 7.4(a) along with Lemma 7.5. Using [LB:Q] = ℓ+ 1 and
(7.11) to bound for M(LU(ζq)/LB) gives
πf(x, a;ℓ)≤πC′×S(x, LU(ζq)/LB)+3.15546ℓ(ℓ+ 1) x1/2
log x
+ (ω(Nf) + 4)(ℓ+ 1) log(ℓ+ 1) + (ℓ+ 1) log(2 rad(Nf)2φ(q) rad(q)) + 1.
(7.24)
If a= 0, we instead use Lemma 7.4(b), and (7.12) to bound M(LH(ζq)/LB). We obtain
a similar bound for πf(x, 0; ℓ), this time with a main term of πC′′ ×S(x, LH(ζq)/LB).
Under the range condition (B), the sum of the error terms above will be less than
0.01 x
φ(q)(ℓ1−1) log x. Thus we have that
πf(x, a;ℓ)≤πC′×S(x, LU(ζq)/LB)+0.01 x
φ(q)(ℓ1−1) log x.(7.25)
The analogous equation holds when a= 0. Since xand ℓ1satisfy the preconditions
of Lemma 7.10, we have
πC′′×S(x, LH(ζq)/LB)≤11.29|S|x
φ(q)(ℓ1−1) log x.(7.26)
If a= 0, we instead use Lemma 7.9. (7.25) and (7.26) then give
πf(x, a;ℓ)≤11.3|S|x
φ(q)(ℓ1−1) log x(7.27)
for all primes ℓsatisfying conditions (A) and (B) of the theorem statement.
Let ℓ1< ℓ2<··· < ℓtbe the tprimes determined in condition (D) of the theorem
statement. Due to (C) and (D), the preconditions of Corollary 7.2 hold so we have
πf(x, a)≤2r
log 2 log log x+ 1max
1≤j≤tπf(x, a;ℓj)
+x√2
(log x)r+2x1/2
(log x)r+ 2(log x)2r+ 35.(7.28)
Therefore, using that x≥e272 from (7.13), we can combine error terms to obtain that
πf(x, a)≤2r
log 2 +1
log 272log log xmax
1≤j≤tπf(x, a;ℓj)+ 1.42 x
(log x)r.(7.29)
Finally, we combine (7.29) with the estimates for πf(x, 0; ℓj) provided in (7.27), and
in order to take the maximum we note that the right-hand side of (7.27) is maximized
over the ℓj’s when j= 1. □
In the next sections, we make Theorem 7.11 explicit in two different cases, proving
Theorems 1.2 and 1.3.
Remark 7.1. Theorem 7.11 can be made explicit for other non-CM cusp forms fif
the surjectivity of the ℓ-adic representation ρf,ℓ is known beyond some threshold ℓ > ℓ0.
Explicit values of ℓ0are only known for certain forms, such as the newforms of level 1
and weights k∈ {12,16,18,20,22,26}.

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 33
We also require bounds for the number of primes in the interval [ℓ, 2ℓ] such that
gcd(ℓ−1, kf−1) = 1; we supply such bounds for kf= 12 in Lemma 8.3.
8. Proof of Theorems 1.2 and 1.4
We first specialize to the delta function f(z) = ∆(z) defined in (1.12). Recall that ∆ is
a cuspidal newform of weight k∆= 12 and level N∆= 1 with its n-th Fourier coefficient
equal to Ramanujan’s tau-function τ(n). By work of Swinnerton-Dyer [SD73, Theorem
4, Corollary (i)], if ℓ > 691 is prime, then the associated residual representation ρ∆,ℓ is
surjective. Since N∆has no prime factors, we have that ω(N∆) = 0 and rad(N∆) = 1.
Moreover, Serre [Ser73] proved that if τ(p) = 0, then plies in one of |S|= 33 congruence
classes modulo
q:= 3488033912832000
= 214 ×37×53×72×23 ×691.
We apply this data to the statement of Theorem 7.11. First, we require bounds for
functions which count rational primes.
8.1. Auxiliary results on bounds for prime-counting functions. We first state a
version of the classical Brun–Titchmarsh theorem due to Montgomery–Vaughan [MV73]:
Theorem 8.1. Let xand ybe positive real numbers, and let dand qbe relatively prime
positive integers. If π(x;q, d)=#{p≤x|p≡d(mod q)}, then
π(x+y;q, d)−π(x;q, d)<2y
φ(q) log(y/q).
We also need an estimate for the number of primes between yand 2yfrom [RS62].
Lemma 8.2 (Rossen–Schoenfeld [RS62, Corollary 3]).For all y≥20.5, we have
π(2y)−π(y)>0.6y
log y(8.1)
Combining these results gives the following lemma, which we use to verify conditions
(C) and (D) of Theorem 7.11.
Lemma 8.3. Let y≥2000. Then
#{p∈(y, 2y]|p≡ 1 (mod 11)}>0.3y
log y.(8.2)
Proof. The count on the left-hand side of (8.2) is equal to
π(2y)−π(y)−(π(2y; 11,1) −π(y; 11,1)).
By Theorem 8.1 and the assumption that y≥2000, we bound π(2y; 11,1) −π(y; 11,1)
by
π(2y; 11,1) −π(y; 11,1) ≤0.2y
log y−log 11 ≤0.3y
log y.(8.3)
Combining this estimate with Lemma 8.2 gives
π(2y)−π(y)−(π(2y; 11,1) −π(y; 11,1)) >(0.6−0.3) y
log y= 0.3y
log y(8.4)
as desired. □

34 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
8.2. Proof of Theorem 1.2.We apply Theorem 7.11, choosing ℓ1and mso that the
conditions of the theorem hold. For general a∈Z, we take q= 1 and recall that Nf= 1
so that the range condition (B) becomes
x≥e42e62(ℓ+1)(ℓ+ 1)18.9(ℓ+1).(8.5)
If a= 0, we instead use q= 3488033912832000 and the alternate range in condition
(B), so we have
x≥e156e252(ℓ+1)(ℓ+ 1)14.7(ℓ+1).(8.6)
Letting θ > 0 be some fixed parameter to be specified shortly, we will choose the prime
ℓ1to be near (but smaller than) the function
ℓ(x):=θlog x
log log x.(8.7)
We take θ= 0.06 for general a∈Z, and θ= 0.04 when a= 0. When x≥ee16 , we can
verify that (8.5) holds with ℓ=ℓ(x) and θ= 0.06 and that (8.5) holds with θ= 0.04.
We can also verify that condition (C) holds.
More precisely, we set ℓ1to be the largest prime number less than ℓ(x) which is not
congruent to 1 modulo 11. By Lemma 8.3, we can take ℓ1>0.5ℓ(x). Now, setting
r= 4, we use Lemma 8.3 to verify that condition (C) holds when x≥ee16 . Now, using
θ= 0.06, r= 4, and q= 1 gives
π∆(x, a)≤4626.7x(log log x)2
(log x)2+ 1.42 x
(log x)4.(8.8)
Using θ= 0.04, r= 4, and q= 3488033912832000 likewise gives
π∆(x, 0) ≤3.007 ·10−10 x(log log x)2
(log x)2+ 1.42 x
(log x)4.(8.9)
If x≥ee16 , then
1.42 x
(log x)4≤10−16 x(log log x)2
(log x)2.(8.10)
Absorbing the error term into the main term yields the desired result. □
Next, we use Theorem 1.2 to prove Theorem 1.4.
8.3. Proof of Theorem 1.4.We compute a lower bound for
D∆= lim
x→∞
#{1≤n≤x|τ(n)= 0}
x,
using our result for π∆(x, 0). These two quantities are related by the following result.
Proposition 8.4. For all X0≥2, we have that
D∆=Y
τ(p)=0 1−1
p+ 1(8.11)
>exp −Z∞
X0
π∆(x, 0)
x(x+ 1) dx·Y
p≤X0
τ(p)=0 1−1
p+ 1.(8.12)

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 35
Proof. The first line follows from [Ser81, Equation 202]. The second line follows from
a partial summation argument applied to the logarithm of the product in (8.11), and
the identity for the truncated integral
exp −ZX0
2
π∆(x, 0)
x(x+ 1) dx=1 + 1
X0mm
Y
j=1 1−1
pj+ 1
where p1, . . . , pjare the primes less than X0with τ(p) = 0. □
Proof of Theorem 1.4.We use Proposition 8.4 with the cutoff X0= 1023. By computer
search, Rouse and Thorner [RT17] showed that all but 1810 prime numbers pless
than 1023 satisfy τ(p)= 0, using Swinnerton-Dyer’s congruences for τ(n) and Galois
representations at ℓ= 11,13,17,19, which are necessary conditions for τ(p) = 0 [SD73].
They compute
Y
τ(p)=0
p≤1023 1−1
p+ 1>0.99999999999999999980399.(8.13)
From the fact that if τ(p) = 0, then plies in one of 33 possible residue classes modulo
q, Theorem 8.1 allows us to bound
π∆(x, 0) ≤1810 + 2(x−1023 + 2q)
φ(q) log((x−1023 + 2q)/q)×33, x > 1023.(8.14)
To bound the contribution in this range, we use the following lemma.
Lemma 8.5. If X0>2q > 0, then
ZX1
X0
x−X0+ 2q
x(x+ 1) log((x−X0+ 2q)/q)dx < ZX1/q
2
x
(x+X0/q −2)2log xdx.
Proof. We use that X0−2q > 0, x(x+ 1) > x2when x > 0, and set u=x−X0+ 2q
to find that
ZX1
X0
x−X0+ 2q
x(x+ 1) log((x−X0+ 2q)/q)dx < ZX1
2q
u
(u+X0−2q)2log(u/q)du.
Now, setting t=u/q and simplifying completes the proof. □
We want to bound the integral in (8.12) between X0= 1023 and X1=ee16. (8.14)
and Lemma 8.5 allow us to numerically integrate to find that
ZX1
X0
π∆(x, 0)
x(x+ 1) dx < ZX1
X0
1810
x(x+ 1) dx +66
φ(q)ZX1/q
2
x
(x+X0/q −2)2log xdx
≤1.1358 ×10−12.

36 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
In the range x>X1, we upper bound π∆(x, 0) using Theorem 1.2, which gives
Z∞
X1
π∆(x, 0)
x(x+ 1) dx ≤(3.01 ·10−10 )Z∞
X1
(log log x)2
x(log x)2dx
= (3.01 ·10−10)·(log log X1)2+ 2 log log X1+ 2
log X1
<9.824 ·10−15.
Thus we have that
exp −Z∞
X0
π∆(x, 0)
x(x+ 1) dx>exp(−1.1358 ×10−12 −9.824 ·10−15 )
>0.999999999998854.(8.15)
Combining the estimates (8.13) and (8.15) via Proposition 8.4 yields
D∆>0.99999999999999999980399 ·0.999999999998854
>0.99999999999885
= 1 −1.15 ·10−12 ,
as desired. □
9. Proof of Theorems 1.3 and 1.5
Let Ebe the elliptic curve defined by (1.21). We apply our method from Section 8
to the modular form fE(z) associated to E, as defined in (1.22), in order to bound
πE(x, a) and DfE. We first note the following congruence relation.
Lemma 9.1. If p= 11, then aE(p)≡p+ 1 mod 5.
Proof. The elliptic curve Esatisfies E[5] ∼
=Z/5Z[LT76, p. 55]. If p= 11, then Ehas
good reduction at p, so the reduction mod pmap E[5] →E(Fp) is injective [Sil09,
Proposition VII.3.1]. Then, #E(Fp)≡0 mod 5, and by the modularity theorem
aE(p) = p+ 1 −#E(Fp), so we have that p+ 1 −aE(p)≡0 mod 5. □
This allows us to set q= 5 below. Note that if a≡1 mod 5 and aE(p) = a, then
p= 5 or 11. If a= 1 and a≡1 mod 5, then πE(x, a) = 0 since aE(5) = aE(11) = 1.
For other values of a, we use Theorem 1.3.
Proof of Theorem 1.3.We use Theorem 7.11 as in the proof of Theorem 1.2 in Sec-
tion 8.2. Substituting q= 5 and Nf= 11 into (B) yields the range condition
x≥e72e46(ℓ+1)(ℓ+ 1)18.9(ℓ+1).(9.1)
We repeat the arguments of Section 8.2, verifying that the conditions of Theorem 7.11
hold with θ= 0.055, r= 2 and x≥ee13 . This implies that
πE(x, a)≤630.91 ·x(log log x)2
(log x)2+ 1.42 ·x
(log x)2.(9.2)

AN EXPLICIT CHEBOTAREV VARIANT OF THE BRUN–TITCHMARSH THEOREM 37
If x≥ee13 , then
1.42 x
(log x)2≤0.01 ·x(log log x)2
(log x)2.(9.3)
Substituting (9.2) into (9.3) completes the proof. □
We now prove Theorem 1.5.
Proof of Theorem 1.5.We argue as in the proof of Theorem 1.4 in Section 8.3. By a
similar argument as in the proof of Proposition 8.4 (applying [Ser81, Equation 201]
instead of [Ser81, Equation 202]), we first note that
DfE=14
15 Y
aE(p)=0 1−1
p+ 1
>14
15 exp −Z∞
X0
πE(x, 0)
x(x+ 1) dx·Y
p≤X0
aE(p)=0 1−1
p+ 1.(9.4)
We use the cutoff X0= 1011. By computer search, Rouse and Thorner [RT17] find that
there are precisely 17857 primes p≤1011 such that aE(p) = 0, and they compute that
14
15 Y
p≤1011
aE(p)=0 1−1
p+ 1= 0.8465247961... (9.5)
If x > 1011, then Lemma 9.1 and Theorem 8.1 imply that
πE(x, 0) ≤17857 + 2(x−1011 + 10)
4 log((x−1011 + 10)/5).(9.6)
We want to bound the integral in (9.4) between X0= 1011 and X1=ee13. Applying
(9.6), Lemma 8.5 and integrating numerically gives the bound
ZX1
X0
17857 + 2(x−1011+10)
4 log((x−1011+10)/5)
x(x+ 1) dx < 4.898.
When x > X1, we upper bound πE(x, 0) using Theorem 1.3, which yields
Z∞
X1
πE(x, 0)
x2+xdx ≤631 Z∞
X1
(log log x)2
x(log x)2dx < 0.281.
We then have that
exp −Z∞
1011
πE(x, 0)
x(x+ 1) dx>exp(−4.898 −0.281) >0.00563364.(9.7)
Combining (9.5) and (9.7) via (9.4) completes the proof. □

38 DANIEL HU, HARI R. IYER, AND ALEXANDER SHASHKOV
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Department of Mathematics, Princeton University
Email address:danielhu@princeton.edu
Department of Mathematics, Harvard University
Email address:hiyer@college.harvard.edu
Department of Mathematics, Williams College
Email address:aes7@williams.edu