November 21, 2022
A MODULAR ANALOGUE OF A PROBLEM OF
VINOGRADOV
R. ACHARYA, S. DRAPPEAU, S. GANGULY, AND O. RAMARÉ
Abstract. Given a primitive, non-CM, holomorphic cusp form f with normalized Fourier coefficients apnq and given an interval I Ă r´2, 2s, we study
the least prime p such that appq P I . This can be viewed as a modular form
analogue of Vinogradov’s problem on the least quadratic non-residue. We obtain strong explicit bounds on p, depending on the analytic conductor of f forsome specific choices of I.

1. Introduction
The present article is concerned with understanding the distribution of theinitial Fourier coefficients of primitive holomorphic cusp forms at primes. Suppose
f is such a form of weight k for the group Γ0 pN q. We further assume that f is
non-CM and has trivial nebentypus. The normalized Fourier coefficients of f at
infinity are denoted by papnqqně1 , so that ap1q “ 1 andf pzq “8ÿk´1

apnqn 2 epnzq,

n“1where, as usual, epzq denotes e2πiz and with this normalization, the Ramanujan
bound (proved by Deligne [6]) says ´2 ď appq ď 2 for primes p. Furthermore,
the function n ÞÑ apnq is real-valued and multiplicative. We refer the reader tothe text [12] for background information on holomorphic modular forms. The
Sato-Tate conjecture for distribution of the angles θp , defined by appq “ 2 cos θp ,as p runs over primes, which is now a theorem of Clozel, Harris, Shepherd-Barron
and Taylor [3, 31, 9], implies, in particular, that any interval of positive measure
within r´2, 2s contains infinitely many values of appq. The goal of this article
is to obtain bounds for the least prime p such that appq lies in a fixed intervalI Ă r´2, 2s. This can be considered as an analogue of Vinogradov’s problem of
estimating, given a modulus q ě 1, the size of the least quadratic non-residue
modulo q (see [2], [32]). The quality of our bounds will be measured in terms of
the analytic conductor qpf q “ N k 2 of the form f (see §2.1), and also separately interm of the weight k of the form, considering the level N to be fixed and in termsof the level N , considering the weight k to be fixed. We restrict our attention toforms with trivial nebentypus in order to clarify the presentation but the methods
presented here can be extended to a more general setting.
Let I Ă r´2, 2s. Theorem 1.6 of the paper [22] of Lemke-Oliver and Thorner
implies that there exists a constant A depending only on I such that appq P I forsome prime p ď q A . Their method relies on effective log-free zero density estimates
for the L-function associated with f , and the Turán power-sum method. The
value of the constant A is not stated explicitly in their paper but it is not hard
to see that the constant is effective and can be worked out explicitly. However2010 Mathematics Subject Classification. 11M06, 11N56, 11N80.
Key words and phrases. Linnik’s theorem, modular forms, Moebius function, Sato-Tate law.
1

2

R. ACHARYA, S. DRAPPEAU, S. GANGULY, AND O. RAMARÉ

the method is likely to produce quite large values of A. Our aim in the present
work is to make the value of A as small as possible for some specific intervals.
We define, when κ is positive and x P r0, 1s:ż x{p1`xq κ´1
ÿ 1 ´ x ¯κ`kh dh
F px; κq “
“
.
(1.1)
1´h
κ`k 1`x0
kě0Note that F p¨; κq is increasing between F p0; κq “ 0 and F p1; κq “
We thus define a function G p¨; κq with value in r0, 1s by
G py; κq “ maxtx P r0, 1s : F px; κq ď 1{yu.

ş1{2 hκ´1 dh
0

1´h

.

(1.2)

The function G is non-increasing and we have G py; κq “ 1 when y ď 1{F p1; κqand by convention G p8; κq “ 0.
We now state our main results which depend crucially on knowledge about theanalytic properties of the symmetric power L functions associated to f (see §2.1
for definition). This is likely to change in the future; only small changes would be
required in our proofs to reflect any such improvement. Here is the assumption
we rely on.
Hypothesis H` : The L-function Lps, sym` pf qq has analytic continuation to
the entire complex plane and it satisfies the boundLp1{2 ` it, sym` pf qq !ε qpsym` pf q, sqλ` `εfor any ε ą 0.
For holomorphic forms, the automorphy of Lps, sym` f q has been known for` ď 8 by [7, 17, 18, 16, 4, 5], and has recently been proved for all ` when N issquarefree by Newton-Thorne [28]. As a result, these L-functions admit holomorphic continuation to the entire complex plane and by the convexity principle, H`
holds with λ` “ 1{4 (known as the convexity bound) for ` ď 8 unconditionallyand for all ` when N is squarefree.
Our results are the following.
Theorem 1.1. For any δ P p0, 2s, let θ1 pδq “ G p2 ` δ; δq. The function θ1 isincreasing and we have θ1 p0`q “ 0 and θ1 p1q “ 0.3956 ¨ ¨ ¨ . Suppose λ1 ą 0 is anexponent that satisfies the hypothesis H` below for ` “ 1, and let ε ą 0. Then for
q “ N or k 2 sufficiently large, there exists a prime
2λ1

p !ε q 1`θ1 pδq`εwith appq ď δ.
Remark 1.2. The convexity bound (Phragmén-Lindelöf principle) allows taking
λ1 “ 14 but better exponents, called subconvex exponents are known in both theweight and the level aspects. For example, one may take λ1 “ 16 when N “ 1 bya result of Jutila and Motohashi [15].
Theorem 1.3. For any δ P p0, 1s, let θ2 pδq “ G pp1 ` δq2 ; 2δ ` δ 2 q. The function
θ2 is increasing when δ ď 0.5305 ¨ ¨ ¨ , and constant equal to 1 afterwards. We have
θ2 p0`q “ 0, θ2 p1{2q “ 0.9093 ¨ ¨ ¨ , θ2 p1q “ 1. Suppose λ2 ą 0 is an exponent thatsatisfies the hypothesis H` below for ` “ 2, and let ε ą 0. For any δ P r0, 1s, andfor q “ N or k 2 sufficiently large, there exists a prime
4λ2

p !ε q 1`θ2 pδqwith |appq| ď 1 ` δ.

`εA MODULAR ANALOGUE

3

Remark 1.4. The convexity bound allows the choice λ2 “ 14 and currently thisis the best known exponent. Obtaining a subconvex estimate for the symmetricsquare L-function in the level or the weight aspect is a challenging problem.
It turns out that showing the existence of primes p of small size in terms ofthe conductor (i.e., weight and level) such that appq ě 0 is rather difficult. Byutilizing the fact that hypotheses H` holds true for 1 ď ` ď 5, we are able toshow the following result:
Theorem 1.5. There is a prime p ! k 24 N 21 such that appq ě 0.
The results above are all obtained using a similar strategy and this is summarized in Theorem 1.11 below. For some specific intervals, however, we obtainbetter bounds by employing ad hoc techniques using L-functions as we now describe.
Theorem 1.6. For any ε ą 0, there is a prime p “ Oε pkN q1`ε such thatappq ă 0.
Corollary. The least prime such that appq ‰ 0 is !ε pkN q1`ε , for any ε ą 0.
Remark 1.7. As the proof of the above theorem shows, the exponent 1 can be
replaced by 4λ2 and any subconvex estimate λ2 ă 1{4 for the symmetric square
L-function will lead to an improvement of the above result.
The next result relates the possibility of the initial coefficients at primes assuming extreme values with the size of Lp1, f q. For q “ N k 2 , letγ ´ :“ lim inf
qÑ8log Lp1, f q,
log log qγ ` :“ lim sup
qÑ8log Lp1, f q.
log log q

From the zero-free region of Lps, f q (See [11]), the standard techniques yield´2 ď γ ´ ď γ ` ď 2.

(1.3)

Theorem 1.8. For any δ, ε ą 0, the least prime p such that appq ą γ ´ ´ δ isOpq ε q. Similarly, the least prime p such that appq ă γ ` ` δ is Opq ε q.
Remark 1.9. The bounds (1.3) seem to be the best known, and any improvement
would yield a non-trivial result in Theorem 1.8. The quality of the upper-boundon p, namely Opq ε q, compared to the above results, suggests that improving thebounds (1.3) is a difficult task. Under the Riemann Hypothesis for Lps, f q, one
has the bounds
plog log qq´2 ! Lp1, f q ! plog log qq2 ,at least in the case N “ 1 (see [23, Thm. 3] for a precise and stronger statement),
which yields conjecturally γ ´ “ γ ` “ 0. Furthermore, it is known that thesebounds hold for almost all forms (see [24, Cor. 2] for a precise statement).
Several authors investigated the smallest integer n such that apnq ă 0, see for
instance [13], [19], [21] or [25]. It follows from [25] that the least such n is Opq 3{8 q,where q “ N k 2 . A closer scrutiny of their proofs reveals that the integer n they
produce is either a prime or the square of a prime. Indeed, all the above works
make use of the contrast between the sizes of appq and app2 q forced by the Hecke
relation appq2 ´ 1 “ app2 q for primes p. Since we aim at localizing only appq’s,
the coefficients at primes, we cannot rely on such procedures. In fact, the two
methods we propose are reverse: from a localization on appq, we show that somepolynomial in appq has to be large for many primes p. This polynomial defines the4

R. ACHARYA, S. DRAPPEAU, S. GANGULY, AND O. RAMARÉ

value at p of a new function whose Dirichlet series we approximate with productsof Lps, sym` f q and it is by using the analytic properties of these latter that wereach a contradiction. To find an integer n such that apnq ă 0, only the analytic
properties of Lps, f q are required.
Regarding bounds conditional on the Riemann Hypothesis, Ankeny [1] hasproved that for any non-trivial character χ mod q, if the Riemann hypothesis
is true for Lps, χq, then the least n such that χpnq ‰ 1 is Opplog qq2 q. It is notdifficult to show that the analogous phenomenon holds in our setting:
Theorem 1.10. Assume that for all ` ě 1, the function Lps, sym` f q is entire
and satisfies the Riemann hypothesis. Then for any interval I Ď r´2, 2s of positive
measure, the least prime p such that appq P I satisfies p !I plog qq2 .
Let us now state our general theorem depending on the hypothesis H` . Notethat this result implies Theorems 1.1, 1.3 and 1.5.
Theorem 1.11 (Generic theorem). Let pb` q1ď`ďL be non-negative integers, Letκ ą 0 and F be real, and let I Ă r´2, 2s be such that
ÿ
$
@x
P
r´2,
2s
r
I,
b` U` px{2q ě κ ą 0,
’
’
&
1ď`ďL
ÿ
(1.4)
’
b` U` px{2q ě F,’%@x P r´2, 2s,
1ď`ďL

where U` are the Chebyshev polynomials of the second kind. Then, on assumingpH` q`ďL , the least prime p such that appq P I satisfiesř
2 ` `b` λ`
log p
ď
` ε,
(1.5)
log N
1 ` G pκ ´ F ; κq
for any ε ą 0 and N large enough with respect to the weight k and ε; andř
2 ` p` ` p`qqb` λ`
log p
ď
` ε.
(1.6)
log k
1 ` G pκ ´ F ; κq
for any ε ą 0 and k large enough with respect to the level N and ε. Here p`q “1´p´1q`
P t0, 1u is the parity of `.
2
The intervals rα, βs for which there is a linear combination with non-negative
coefficients of U1 , . . . , U8 which takes positive values outside rα, βs delimit acurve in pα, βq, whose exact determination is an interesting question (without
the non-negativity condition, the analogue for U1 , . . . , U4 was solved in Appendix A of [22]). Between this curve and the diagonal α “ β, Theorem 1.11 yieldspan upper-bound on log
log q , which gets smaller as one moves away from the diagonal.
This is represented in Figureř1, which was obtained by case-by-case analysis of
all linear combinations with `ď8 `b` ď 42. On the left, darker colors indicate alarger upper-bound.
Theorem 1.11 should be compared with Theorem 1.8 of [22]. In both cases,
we are given an interval I Ă r´2, 2s, and we are looking for the least prime p suchthat appq P I. In Theorem 1.8 of [22], the authors obtain an exponent dependingon the quality with which the indicator function 1I can be minorized by a linear
combination of U0 , U1 , U2 , . . . . In Theorem 1.11, we obtain an exponent depending on the quality with which the complementary indicator function 1r´2,2srI is
minorized by a linear combination with non-negative coefficients of U1 , U2 , . . . . An
inconvenient of our method is that there is no clear description of the allowableA MODULAR ANALOGUE

5

log p
Figure 1. Upper-bound on log
N in Theorem 1.11 for I “ rα, βs.

intervals I. Theorems 1.1-1.5 indicate that, when it can be applied, the method
described here yields non-trivial numerical results.
Notation. Our notation is quite standard. We follow the usual practice of denoting by p an arbitrary prime and by ε an arbitrarily small positive real number
which need not be the same in every occurrence. For any set X Ă R and maps
F : X ÞÑ C and G : X ÞÑ r0, 8q, we write
F pxq ! Gpxq or F pxq “ OpGpxqq
if there exists a C ą 0 such that |F pxq| ď CGpxq for all x P X. Sometimes, theimplied constant C depends on some parameters and this dependence is shown inthe subscript. For example, often the implied constant depends on the parameterε, an arbitrarily small positive real number and we display this dependence by
writing !ε or Oε . Sometimes, the dependence is not shown when it is clear fromthe context in order to avoid making the notation too cumbersome. By η̌, wedenote the Mellin transform of a function η:
ż8η̌psq “ηptqts´1 dt.
(1.7)
0

Acknowledgements
The first author would like to thank Ramakrishna Mission Vivekananda Educational and Research Institute for a wonderful academic atmosphere and the
National Board for Higher Mathematics (DAE file reference no. 0204/12/2018/R& D II/ 6463) for financial support.
The third author thanks CNRS (Centre National de la Recherche Scientifique)through its Laboratoire International Associé “Indo-French Program for Mathematics” and ISI (Indian Statistical Institute) for financial support that enabled
him to visit Aix-Marseille Université in June, 2017, where this work was initiated.
He also thanks the department of Mathematics of Aix-Marseille Université for its
warm hospitality.

6

R. ACHARYA, S. DRAPPEAU, S. GANGULY, AND O. RAMARÉ

2. Background on modular forms and L-functions
2.1. Symmetric power L-functions. For a primitive form f , as in the introduction, its normalized coefficients af ppq “ appq can be written asappq “ αf ppq ` βf ppq
where, for p - N , αf ppq “ 1{βf ppq and both are complex numbers of absolutevalue 1. For each ` P N, the `-th symmetric power L-function of f is defined, for<s ą 1, byź ź `ÿ asym` f pnq
˘´1
Lps, sym` f q “
1 ´ αf ppq`´j βf ppqj {ps
“:
.
(2.1)
ns
p 0ďjď`ně1
We have sym1 f “ f and it is convenient to set sym0 f “ 1 so that Lps, sym0 f q “ζpsq. It is expected from a general conjecture of Langlands [20] that for every `,there is a cuspidal automorphic representation of GL``1 pAQ q that corresponds to
the L-function Lps, sym` f q. For 1 ď ` ď 8, this was shown in [7] (for ` “ 2), [17](for ` “ 3), [18, 16] (for ` “ 4) and [4, 5] (for 5 ď ` ď 8). When N is squarefree,
this has been announced for all ` ě 0 in [28].
Following [14, Eq.(5.5)]), we define the analytic conductor of Lps, sym` f q as
qps, sym` pf qq “ N ` p|t| ` 2q``1 k ``p`q ,

(2.2)

`

with p`q “ 1´p´1qbeing 1 or 0 according as ` is odd or even, as in the statement
2of Theorem 1.11.
Once we know that a symmetric power L-function comes form an automorphic representation, the analytic continuation and functional equation for that Lfunctions follows from [8] and thus the Phragmén-Lindelöf convexity principle (orthe approximate functional equation [14, eq. (5.20)]) implies that for 1 ď ` ď 8,
the hypothesis H` holds with the value λ` “ 1{4, even for δ “ 0. This is known
as the convexity bound. Giving a bound on an L-function that is stronger thanthe convexity bound is a challenging problem which has been solved in a few
cases (see [27] and the references therein) and this is known as the subconvexity
problem. Sometimes we are interested in the size of the L-functions in terms of
only the size of the variable t, or the weight k or the level N . A result of Jutila
and Motohashi [15] says that taking λ1 “ 1{6 is permissible in the weight andthe t-aspect. We further define
qpsym` pf qq :“ N ` k ``p`q .

(2.3)

In particular, qpf q “ N k 2 and qpsym2 pf qq “ N 2 k 2 . Note that in the weightaspect, qpf q and qpsym2 pf qq are of the same order.
For the coefficients of the symmetric `-th power L-function of f , we have thefollowing relation for every prime p:` ˘sinpp` ` 1qθppqq
asym` f ppq “ a p` “ U` pcos θppqq “ U` pappq{2q “,(2.4)
sin θppq
where U` is the Chebyshev polynomial of second kind, whose properties we recall
next.
2.2. Chebyshev polynomials of the second kind. We recall that the Chebyshev polynomial of second kind pU` q`ě0 are defined by
U0 “ 1,U1 “ 2x,U``1 ´ 2xU` ` U`´1 “ 0.

(2.5)

A MODULAR ANALOGUE

7

These polynomials form an orthonormal basis in the space of polynomials on theinterval r´1, 1s relative to the Hermitian productż1axf, gy “
f pxqgpxq π2 1 ´ x2 dx.
(2.6)
´1

The first few are given by

U2 “ 4x2 ´ 1,
U3 “ 8x3 ´ 4x,
U4 “ 16x4 ´ 12x2 ` 1,
U5 “ 32x5 ´ 32x3 ` 6x,
U6 “ 64x6 ´ 80x4 ` 24x2 ´ 1,
U7 “ 128x7 ´ 192x5 ` 80x3 ´ 8x,
U8 “ 256x8 ´ 448x6 ` 240x4 ´ 40x2 ` 1.
The last equality in Eq. (2.4) comes from the relation
Un pcos θq “sinppn ` 1qθq.
sin θ

3. Auxiliary Lemmas
3.1. Convolutions.
Lemma 3.1. Assume pH` q1ď`ďL . Let L ě 1 be an integer and let pb` q0ď`ďL be acollection of non-negative integers. Then, we have the equality
ř
˙
`
źˆ
ź
` b` app q
1`“
Lps, sym` f qb` Hpsq,
s
p
p
0ď`ďL
where H is a function that is holomorphic and bounded by a constant in the region<s ě 12 ` ε for any ε ą 0.
Proof. This follows easily by comparing the p-th Euler factors.

We recall that, in the half-plane of absolute convergence, we have
˙
ˆ
˙ ˆ
˙
źˆ
appq
1 ´1 ź
αppq ´1
βppq ´1
Lps, f q “
1 ´ s ` 2s
“
1´ s
1´ s
p
p
p
p
p
p

(3.1)

as well as
2

Lps, sym f q “źˆpappq2 ´ 1 appq2 ´ 1
1
1´
`
´ 3s
s
2s
p
p
p

˙´1
.

(3.2)

3.2. Averages of multiplicative functions. We quote Theorem 21.2 of [29]which follows an idea of Wirsing [33].
Lemma 3.2. Let f be a non-negative multiplicative function and κ be a nonnegative real parameter such that` ν˘ ` ν˘$ ÿfp log p “ κQ ` OpQ{ logp2Qqq
pQ ě 1q,’’’’& pě2,νě1pν ďQÿÿ` ˘ ` ˘ ` ˘ a’f p` f pν log pν ! Q,’’’
% pě2 ν,`ě1,pν`` ďQ

8

R. ACHARYA, S. DRAPPEAU, S. GANGULY, AND O. RAMARÉ

then we haveÿf pdq “ κ C ¨ D plog Dqκ´1 p1 ` op1qq,

dďD

where

˙
*
ź"ˆ
1
1 κ ÿ ` ν˘
C“
1´
f p
.
Γpκ ` 1q p
p
νě0

(3.3)

Lemma 3.3. Under the same hypotheses
ş1 of Lemma 3.2 we have, for any continuously differentiable function η with 0 ηpuqdu ‰ 0:żDÿ
f pdqηpd{Dq “ κCp1 ` op1qq
plog uqκ´1 ηpu{Dqdu2dďDas D ÝÑ 8.
The condition on η is obviously satisfied if, as will be the case for us, η isnon-negative with support inside the interval r0, 1s.
Proof. Using Lemma 3.2, we find thatż1ÿÿ
ÿf pdqηpd{Dq “
f pdqηp1q ´
f pdq
η 1 ptqdt
dďD

dďD

ÿ

f pdqηp1q ´

“

dďD
ż1 ÿ

d{D

f pdqη 1 ptqdt

0 dďtDdďD“ κηp1qC ¨ D plog Dqκ´1żD´κCuplog uqκ´1 η 1 pu{Dqdu{D ` opDplog Dqκ´1 q
2

as D Ñ 8. Hence, by partial summation,żDÿ´1
f pdqηpd{DqpκCq “pκ ´ 1 ` log uqplog uqκ´2 ηpu{Dqdu ` opDplog Dqκ´1 q
2

dďD

żD
plog uqκ´1 ηpu{Dqdu ` opDplog Dqκ´1 q.

“
2However we also haveżD´ ż D{ log D
¯ żD
κ´1
κ´1
plog uq ηpu{Dqdu “ Oplog uq du `22

plog uqκ´1 ηpu{Dqdu

D{ log D
ż1
κ´1

`
˘
“ O Dplog Dqκ´2 ` Dplog Dq1{ log D´1`

log v ¯κ´1
ηpvqdv
log D

ż1
`
˘
“ O Dplog Dqκ´2 ` Dplog Dqκ´1

ηpvqdv1{ log D´ż 1„¯ηpvqdv Dplog Dqκ´10

as D Ñ 8, since 0 ηpuqdu ‰ 0. In the third line we have used the uniform
estimate p1 ` plog vq{ log Dqκ´1 “ 1 ` Oplogp1{vq{ log Dq for 1{ log D ă v ă 1.
Hence our claimed estimateżDÿ
f pdqηpd{Dq “ κCp1 ` op1qq
plog uqκ´1 ηpu{Dqdu
ş1

dďD

2

A MODULAR ANALOGUE

follows.

9
4. A general average boundLemma 4.1. Let L P Ną0 , and assume pH` q1ď`ďL . Let pb`ř
q0ď`ďL be a collectionof non-negative integers. Given a primitive form f pzq “ ně1 apnqepnzq as in
the introduction, let us define a multiplicative function hf by the equalityř˙`ÿ hf pnq źˆ` b` app q
“
1`
ns
ps
n
p
Then hf is supported on square-free integers and there exists a polynomial PL ofdegree at most b0 ´ 1 such that, for any ε ą 0, we have
´ 1
¯
ÿ
ź
hf pnqηpn{Xq “ XPL plog Xq ` O X 2 `εqpsym` pf qqb` λ` `ε
(4.1)
ně1

1ď`ďL

for any compactly supported twice continuously differentiable non-negative function η.
Proof. Let us denote by S the left-hand side of (4.1). By taking Mellin transforms(e.g. p.90 of [14]), we getż 2`i8
ÿ hf pnq
1
S“X s η̌psqs
ds.
2iπ 2´i8
ns
ně1
The fact that η is twice continuously differentiable ensures us that its Mellintransform verifies η̌psq ! 1{p1 ` |s|2 q uniformly in any closed vertical strip intheř half planes<s ą 0. Lemma 3.1 gives us an expression for the Dirichlet series
ně1 hf pnq{n from which we see that we can shift the line of integration to<s “ 12 ` ε obtaining that the error term is at most´ 1¯
źO X 2 `εqpsym` pf qqb` λ` `ε ,1ď`ďLby our hypothesis pH` q1ď`ďL and the convexity principle. The residue at 1 gives
the claimed main term, and the lemma follows readily.

5. A general Lemma around Vinogradov’s trick
Lemma 5.1. Let g be a real-valued multiplicative function supported on thesquarefree integers. We assume further that gppq ě F for every prime p, andthat for every prime p ď P , we have gppq ě κ ą 0. Let η be a non-negative,
ş1
continuously differentiable function with support within r0, 1s such that 0 ηpvqdv “
1. We have, for M “ P θ for some θ P r0, 1s,
ˆ
˙
ÿ
`
˘
n
µ2 pnqgpnqη
ě p1 ` op1qqκ CM P plog M P qκ´1 1 ´ pκ ´ F qF pθ; κqPM
ně1where C is given by (3.3) and F is defined in (1.1)
The factor µ2 pnq is only here to remind the reader that the variable n is
restricted to squarefree values. It can be omitted!
Proof. We set
ˆ

˙
n
S“
gpnqη
.
PM
ně1
ÿ

(5.1)

10

R. ACHARYA, S. DRAPPEAU, S. GANGULY, AND O. RAMARÉ

By our hypotheses, we find that
ˆ
˙
ÿ
ÿ
n
S“
gpnqη
`
gppq
P
M
nďP M,P ăpďP M

ˆ
ÿ

gpnqη

nďP M {ppnPM˙P ` pnqďP

ˆ

n
ě
gpnqη
PM
nďP M,

˙

ÿ

`F

ÿ

ÿ

2

ωpnq

µ pnqκ

P ăpďP M nďP M {p

ˆ

˙

2

ωpnq

pn
η
PM

P ` pnqďP

ˆ

ÿ
ě
nďP M

2

µ pnqκ

ωpnq

n
η
PM

˙

ˆ

` pF ´ κqÿÿµ pnqκ

P ăpďP M nďP M {p

˙
pn
η
.
PM

Here P ` pnq denotes the greatest prime divisor of n. We appeal to Lemma 3.3
with f pnq “ µ2 pnqκωpnq and getż PM
S{pCκq ě p1 ` op1qq

´ u ¯
du
PM
2
´ up ¯ÿ ż P M {p` pF ´ κ ` op1qq
plog uqκ´1 η
du.
PMN ăpďP M 2plog uqκ´1 ηNote that the change of variable vP M “ u shows that
ż PM
ż1
´ u ¯
plog uqκ´1 ηdu “ P M plog P M qκ´1
ηpvqdvp1 ` op1qq.
PM20
We use this estimate with M replaced by M {t and the prime number theorem toinfer that´ up ¯ÿ ż P M {pplog uqκ´1 η
duPM
2N ăpďP M
ż1
ż PM´
P M ¯κ´1 dt“ P M p1 ` op1qq
ηpvqdvlog
tt log t0Nwhile this last integral equals, with the change of variable v “ pP M qh and M “N θ,żMż θ{p1`θq κ´1plog vqκ´1 dv
h dhκ´1“ plog P M q.
1´h
1 vplogpP M q ´ log vq
0ş1
Recall that 0 ηpvqdv “ 1. We thus find thatż θ{p1`θq κ´1
p1 ` op1qqSh dh
ě 1 ` pF ´ κq
κ´1
CκP M plog P M q
1´h
0“ 1 ´ pκ ´ F qF pθ, κq.

6. Proof of Theorems 1.11, 1.1, 1.3, 1.5
Suppose appq R I for every p ď P . Under the assumptions of Theorem 1.11,
let θ P r0, 1s be such that1
ą F pθ; κq;
(6.1)
κ´F

A MODULAR ANALOGUE

11

for instance, we may take θ “ maxpG pκ ´ F ; κq ´ ε, 0q. Consider the sum
ÿS“
hf pnqηpn{P M q
ně1

where M P r1, P s. From the upper and the lower bound of S as given by
Lemma 4.1 and 5.1 respectively and noting that b0 “ 0, we obtain,ź1
pP M q 2 `εqpsym` pf qqb` λ` `ε " P M.
1ď`ďL

Therefore, with M “ P θ for some θ P r0, 1s satisfying (6.1), we have2

P !k Nř` `b` λ` `ε1`θ.

This leads to the estimate (1.5) and the other estimate (1.6) is proved in a similarmanner.
Let us inspect what
ř this gives to us under the convexity bound for λ` “ 1{4.
Since the quantity 2 `ě1 `b` λ` takes all the values that are half-positive integers,
we may inspect the first of them one by one. As we did above, we focus on thelevel N .
řFirst case p1{2q `ě1 `b` “ 1{2.řThis is only possible with the choice b1 “ 1,all other b` ’s being 0. We have 1ď`ďL b` U` px{2q “ x which is positive whenx “ appq ą 0. On assuming appq ě δ when p ď P , we see that we may take κ “ δand F “ ´2 and get, for N ě N0 pεq,
log P
2λ1
ď
` ε.
(6.2)
log N
1 ` G p2 ` δ; δq
Hence Theorem 1.1.
řSecond case p1{2q `ě1 `b` “ 1. This is only possible with the choice b2 “ 1,ř
all other b` ’s being 0. We have 1ď`ďL b` U` px{2q “ x2 ´ 1 which is positive whenx “ appq R r´1, 1s. On assuming |appq| ě 1 ` δ when p ď P , we see that we may
take κ “ 2δ ` δ 2 and F “ ´1 and get, for N ě N0 pεq,
log P
4λ2
ď
` ε.
(6.3)
log N
1 ` G p1 ` 2δ ` δ 2 ; 2δ ` δ 2 q
Hence Theorem 1.3.
Finding non-negative values. Let I “ r0, 2s. A numerical computation foundthe coefficients pb` q0ď`ď5 “ p0, 0, 3, 5, 4, 1q, which satisfy (1.4) with κ ě 1{3and F “ ´10. Then Theorem 1.5 follows from the bounds (1.5) and (1.6).
7. Proof of Theorem 1.6
Let η : R` Ñ R` be smooth, compactly supported and such that 1r0,1s ě η ě

1r1{3,2{3s , let ε ą 0, and consider
T pXq “ÿ

µ2 pnqapnqηpn{XqpX ě 1q.

n

By Lemma 4.1, we get
T pXq ! X 1{2 pk 2 N q1{4`ε .
Suppose that appq ě 0 for all primes p ď X. If the inequalityÿµ2 pnqapnqηpn{Xq ě X 1´εn:apnqě1

(7.1)
(7.2)

12

R. ACHARYA, S. DRAPPEAU, S. GANGULY, AND O. RAMARÉ

holds then we easily have
T pXq ě X 1´ε .
Otherwise, suppose that (7.2) does not hold. We writeÿÿT pXq “µ2 pnqapnqηpn{Xq `µ2 pnqapnqηpn{Xqn:0ďapnqă1ÿn:apnqě1n:0ďapnqă1ÿ“ÿµ2 pnqapnq2 ηpn{Xq `

ě

(7.3)

µ2 pnqapnqηpn{Xqn:apnqě1µ2 pnqapnq2 ηpn{Xq `ÿµ2 pnqapnqp1 ´ apnqqηpn{Xq.

n:apnqě1n

Now the last sum is OpX 1´ε{2 q by Deligne’s bound |appq| ď 2 and the negationof (7.2). The first sum can be handled by Rankin-Selberg method (Lemma 4.1)
and is " Lp1, sym2 f qX ` OpX 1{2 pk 2 N 2 q1{4`ε q. Thus we have, using the lower
bound Lp1, sym2 f q " 1{ logpkN q due to Hoffstein and Lockhart [10],T pXq " X{ logpkN q ` OpX 1{2 pk 2 N 2 q1{4`ε q ` OpX 1´ε{2 q.

(7.4)

One of the equations (7.3) and (7.4) must hold and either, in conjunction withequation (7.1), imply the theorem.
8. Proof of Theorem 1.8
By equation (3) of [30], the Deligne bound |appq| ď 2 and Mertens’ theorem(see [14, Eq. (2.15)]), we haveÿ appqlog Lp1, f q “ Oε p1q `
,
p
pďq ε
and therefore

ÿ 1´
log Lp1, f q ¯appq ´“ Oε p1q.
plog log qpďq εHowever, if we had appq ă γ ´ ´ δ for p ď q ε , then we would also haveÿ 1´
log Lp1, f q ¯δappq ´ď Oε,δ p1q ´ log log q,
p
log
log
q
2
pďq ε
which is a contradiction for q large enough, and therefore there must be a prime p ď
q ε such that appq ě γ´ ´ δ. An identical argument shows the existence of p ď q εsuch that appq ď γ` ` δ.
9. Conditional bounds: proof of Theorem 1.10
By the Stone-Weierstrass theorem, the fact that pU` q forms a basis of RrXs,
and the relation (2.4), we may find L ě 1 and real coefficients b0 , . . . , bL dependingon I, with b0 ą 0, such that
ÿ1

pappq P Iqp1 ´ xp q log p ě

pďx

L
ÿ
`“0

b`

ÿ

asym` f ppqp1 ´ xp q log p.

pďx

By Chebyshev’s estimate, the contribution of the term ` “ 0 isÿb0
p1 ´ xp q log p "I x
pďx

(9.1)

A MODULAR ANALOGUE

13

with an absolute constant. To show that the right-hand side of (9.1) is positive
for some x “ OI pplog qq2 q, it therefore suffices to show that for all integer ` ě 1
and all real x ě 1, we have
ÿasym` f ppqp1 ´ xp q log p “ O` px1{2 log qq.
pďxThis is an immediate consequence of the explicit formula [14, eq. (5.33)] (withan additional smoothing, as in [26, eq. (13.28)]) along with classical zero density
estimates [14, Theorem 5.8].
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Email address: ratnadeepacharya87@gmail.com.
Ramakrishna Mission Vivekananda Educational Research Institute, P.O. Belur
Math, Dist. Howrah West Bengal, India. PIN: 711 202.
Email address: sary-aurelien.drappeau@univ-amu.fr
Institut de Mathématiques de Marseille, U.M.R. 7373, Aix Marseille Université,
Centrale Marseille, CNRS, Campus de Luminy, Case 907, 13288 Marseille Cedex 9,FranceEmail address: sgisical@gmail.com
Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 203
Barrackpore Trunk Road, Kolkata 700108, India
Email address: olivier.ramare@univ-amu.fr
CNRS / Institut de Mathématiques de Marseille, Aix Marseille Université,
U.M.R. 7373, Site Sud, Campus de Luminy, Case 907, 13288 MARSEILLE Cedex 9,France