Erratic behavior of Fourier-coefficients of modular forms in short intervals

0.34 MB·

by  Yuri Bilu

Erratic behavior of Fourier-coefficients of modular
7
forms in short intervals
1
0
2
Yuri F. Bilu1,2, Jean-Marc Deshouillers2,3,
b
e Sanoli Gun1,4, Florian Luca1,5,6
F
5 February 16, 2017
1

]
T
Abstract
N
. Letτ(·)betheclassical Ramanujanτ-functionandletk beapositive
h
integersuchthatτ(n)6=0for1≤n≤k/2. (Thisisknowntobetruefor
at k<1023, and, conjecturally, for all k.) Further, let σ be a permutation
m of the set {1,...,k}. We show that there exist infinitely many positive
[ integers m such that |τ(m+σ(1))|<τ(m+σ(2))|<...<|τ(m+σ(k))|.
WealsoobtainasimilarresultforFourier-coefficientsofgeneralnewforms.

2
v
Contents
5
1
9 1 Introduction 1
1 2 Coefficients of newforms 4
0 3 Sieving 6
1. 4 Avoiding Prime Factors from a Sparse Set 17
0 5 Proof of Theorem 1.2 20
7 6 Proof of Theorem 1.4 24
1
:
v 1 Introduction
i
X
r Throughoutthe article a newform of weight κ and level N means a normalized
a
holomorphic new Hecke eigenform of weight κ for Γ (N) with the trivial char-
0
acter. A non-CM newform is an abbreviation for “newform without Complex
Multiplication”.
1SupportedbytheALGANTprogram.
2SupportedbytheIndo-European ActionMarieCurie(IRSESmoduli).
3SupportedbytheCEFIPRAproject5401-A.
4SupportedbyaSERBgrantandtheDAEnumbertheoryplanproject
5SupportedbygrantCPRR160325161141andbyanA-ratedscientistawardbothfromthe
NRFofSouthAfrica.
6Supportedbythegrantno. 17-02804SoftheCzechGrantingAgency.
1
Let f be a newform and
f(z):= a (n)qn, q =e2πiz
f
n≥1
X
be its Fourier expansion at i . In particular, if f is of weight κ=12 and level
∞
N =1 then
f(z)=∆(z):= τ(n)qn =q (1 qℓ)24,
−
n≥1 ℓ≥1
X Y
where τ(n) is the classical Ramanujan function.
The Fourier-coefficients a (n) are algebraic numbers, all belonging to some
f
totally realnumber field. We fix anembedding of this field to R and, from now
on, view the Fourier-coefficients as real algebraic numbers. In the sequel, we
say “coefficients” instead of “Fourier-coefficients”.
There is quite a few results demonstrating “erratic” behavior of the signs
of τ(n), or, more generally, the coefficients of a general newforms; see, for in-
stance, [5, 6, 9, 11, 12] and the references therein. For instance, Matoma¨ki
and Radziwil l [12] show that the non-zero coefficients are positive and negative
with the same frequency, and determine (up to a multiplicative constant) the
asymptotic frequency of sign changes.
In this paper, we work in the orthogonal direction, and study the behavior
of absolute values of non-zero coefficients. Classical results of Rankin [15, 16]
τ(n)
τ(n)2 x12 and limsup| | =+
| | ≍ n11/2 ∞
n→∞
n≤x
X
implythatthesequence τ(n) isnotultimatelymonotonic;inotherwords,each
| |
of the inequalities
τ(m+1) < τ(m+2), τ(m+2) < τ(m+1)
| | | | | | | |
holdsforinfinitelymanym. Inthisarticleweobtain(asaspecialcaseofamore
general result) a similar statement for more than two consecutive values of τ.
Theorem 1.1. Let k be a positive integer such that
τ(n)=0 (1 n k/2). (1.1)
6 ≤ ≤
Then for every permutation σ of the set 1,...,k , there exist infinitely many
{ }
positive integers m such that
0< τ(m+σ(1)) < τ(m+σ(2)) < < τ(m+σ(k)). (1.2)
| | | | ··· | |
In fact, existence of at least one m satisfying (1.2) implies (1.1), see Theo-
rem 1.4 below; in other words, (1.1) is a necessary and sufficient condition for
(1.2) to happen infinitely often.
It is known [19, Theorem 1.4] that τ(n)=0 when
6
n 982149821766199295999 9 1020.
≤ ≈ ·
2
We also refer to the Corollary 1.2 of the unpublished article [3], which claims
that τ(n)=0 for all n 816212624008487344127999 8 1023.
6 ≤ ≈ ·
According to a famous conjecture of Lehmer, τ(n)=0 for all n. If this
6
conjecture holds true, then Theorem 1.1 applies to all k.
Our principal result is the following general theorem.
Theorem 1.2. Let f ,...,f be non-CM newforms and ν ,...,ν distinct pos-
1 k 1 k
itive integers such that
a (ν )=0 (1 i k). (1.3)
fi i 6 ≤ ≤
Then there exist infinitely many positive integers m such that
0< a (m+ν ) < a (m+ν ) < < a (m+ν ). (1.4)
| f1 1 | | f2 2 | ··· | fk k |
In fact, we prove (see Remark 5.8) that for sufficiently large positive num-
ber x, there are at least cx/(logx)k positive integers m x satisfying (1.4).
≤
Here c>0 depends on f ,...,f ,ν ,...,ν and “sufficiently large” translates
1 k 1 k
as “exceeding a certain quantity depending on f ,...,f ,ν ,...,ν ” .
1 k 1 k
In the most interesting special case when f = =f =f and ν ,...,ν
1 k 1 k
···
is a permutation of 1,...,k, we obtain the following immediate consequence.
Corollary 1.3. Let f be a non-CM newform. Let k be a positive integer such
that
a (n)=0 (1 n k). (1.5)
f
6 ≤ ≤
Then for every permutation σ of the set 1,...,k , there exist infinitely many
{ }
positive integers m such that
0< a (m+σ(1)) < a (m+σ(2)) < < a (m+σ(k)). (1.6)
f f f
| | | | ··· | |
In fact, one can do even better: to give a necessary and sufficient condition
for having (1.6) infinitely often.
Theorem 1.4. Let k be a positive integer. Then for a non-CM newform f the
following three conditions are equivalent.
A. We have
a (n)=0 (1 n k/2).
f
6 ≤ ≤
B. For some positive integer ν we have
a (ν+n)=0 (1 n k).
f
6 ≤ ≤
C. For every permutation σ of the set 1,...,k , there exist infinitely many
{ }
positive integers m such that
0< a (m+σ(1)) < a (m+σ(2)) < < a (m+σ(k)).
f f f
| | | | ··· | |
3
Theorem 1.1 follows from this theorem if we take f =∆.
Techniquesoftheproofsrelyonelementaryarguments,sievemethods(Brun’s
sieve, the Bombieri-VinogradovTheorem), and the truth of the Sato-Tate con-
jecture for non-CM forms.
The article is organizedas follows. In Section 2, we briefly review the prop-
ertiesofthe coefficientsof newformsusedinthe sequel. InSections 3and4,we
obtain two sieving results instrumental for the proofs of Theorems 1.2 and 1.4.
Finally, these theorems are proved in Sections 5 and 6, respectively.
1.1 Conventions
Unless the contrary is stated explicitly:
p (with or without indices) denotes a prime number;
•
κ denotes a positive even integer;
•
i,j,k,ℓ,m (with or without indices) denote positive integers;
•
n (with or without indices) denotes a non-negative integer;
•
d (with or without indices) denotes a square-free positive integer;
•
ε,δ denote real numbers satisfying 0<ε,δ 1/2;
• ≤
x,y,z,t denote real numbers satisfying x,y,z,t 2.
• ≥
2 Coefficients of newforms
Inthissection,welistsomewell-knownpropertiesofthecoefficientsofnewforms
which will be used in the proof of Theorem 1.2.
First of all, the Fourier-coefficients a (n) are multiplicative:
f
a (mn)=a (m)a (n) (m,n 1, gcd(m,n)=1). (2.1)
f f f
≥
Furthermore, the values of a at prime powers satisfy the following recurrence
f
relation
a (pℓ+1)=a (p)a (pℓ) pκ−1a (pℓ−1) (ℓ=1,2,...), (2.2)
f f f f
−
where κ is the weight of f.
Both(2.1)and(2.2)wereconjecturedbyRamanujanwhenf =∆andproved
by Mordell [13]. Proofs can be found in many sources; see, for instance [4,
Proposition 5.8.5.].
A much deeper result is the upper estimate
a (p) 2p(κ−1)/2. (2.3)
f
| |≤
4
It was also conjectured by Ramanujan when f = ∆ and proved by Deligne [2,
Th´eor`eme 8.2]. Equivalently, the polynomial T2 a (p)T +pκ−1 can not have
f
−
distinct real roots. Hence we may write the roots as
α =p(κ−1)/2eiθp, α¯ =p(κ−1)/2e−iθp, (2.4)
p p
with θ [0,π]. If θ =0,π (that is, a (p)= 2p(κ−1)/2) then
p p f
∈ 6 6 ±
αℓ+1 α¯ℓ+1 sin(ℓ+1)θ
a (pℓ)= p − p =pℓ(κ−1)/2 p. (2.5)
f
α α¯ sinθ
p p p
−
We may add for completeness that
(ℓ+1)pℓ(κ−1)/2, θ =0,
a (pℓ)= p (2.6)
f (( 1)ℓ(ℓ+1)pℓ(κ−1)/2, θp =π.
−
Another very deep result is the Sato-Tate conjecture, proved recently by
Barnet-Lamb,Geraghty,Harrisand Taylor[1, TheoremB]. Accordingto it, for
a non-CM newform f, the numbers a (p)/p(κ−1)/2 are equidistributed in the
f
interval [ 2,2] with respect to the measure (2/π)√1 t2dt. This means that
− −
for 2 a b 2, we have
− ≤ ≤ ≤
# p x:a (p)/p(κ−1)/2 [a,b] 2 b
{ ≤ f ∈ } 1 t2dt (2.7)
π(x) → π −
Za
p
as x . Here, π(x) is the number of primes up to x.
→∞
An immediate consequence is the following statement.
Proposition 2.1. Let f be a non-CM newform. Then the following holds.
A. The relativedensity ofthesetofprimes psuchthata (p)/p(κ−1)/2 belongs
f
to a given interval of length 2ε does not exceed ε.
B. In particular, the relative density of primes p such that a (p)=0 or
f
2p(κ−1)/2 is 0.
±
Of course, part B was well known long before the proof of the Sato-Tate
conjecture. See Th´eor`eme 15 in [18, Section 7.2] for a much more general and
quantitatively stronger result.
Equations (2.5) and (2.6) imply that a (pℓ)=0 for some ℓ if and only if
f
θ /π Q (0,1). In fact, one knows the following result.
p
∈ ∩
Proposition 2.2. Let f be a non-CM newform. Then for all but finitely many
primes p we have either a (p) 0, 2p(κ−1)/2 or θ /π / Q.
f p
∈{ ± } ∈
For the proof, see [14, Lemma 2.5].
Onemayremarkthatiff isofweightκ 4thenthisholdsforallpwithout
≥
exceptions, see [14, Proposition 2.4].
5
3 Sieving
Inthisandthesubsequentsections,weestablishtwosievingresultsinstrumental
for the proofs of Theorems 1.2 and 1.4.
Theintegerminthissectionisnotnecessarilypositive;itcanbeanyinteger:
positive, negative or 0. The other conventions made in Subsection 1.1 remain
intact.
3.1 The Sieving Theorem
Let Σ be a finite set of prime numbers. We call m Z
∈
Σ-unit, if all its prime divisors belong to Σ;
•
Σ-square-free, if m is a product of a Σ-unit and a square-free integer.
•
Also, for z 2 we define
≥
P (z)= p. (3.1)
Σ
p<z
pY∈/Σ
Now let a ,...,a ,b ,...,b Z be integers satisfying
1 k 1 k
∈
a =0, gcd(a ,b )=1 (i=1,...,k), (3.2)
i i i
6
a b a b =0 (1 i<j k). (3.3)
i j j i
− 6 ≤ ≤
We consider linear forms L (n)=a n+b and for x z 2, we set
i i i
≥ ≥
✵(x,z)= n:0 n x, gcd L (n) L (n), P (z) =1 . (3.4)
1 k Σ
≤ ≤ ···
Finally, we deno(cid:8)te by ✵ (x,z), the su(cid:0)bset of ✵(x,z) consisti(cid:1)ng of(cid:9)n for which
1
L (n),...,L (n) are Σ-square-free composite numbers.
1 k
The principal result of this section is the following theorem.
Theorem 3.1. Assumethat Σ contains all primes p 2k, all prime divisors of
≤
every a , and all prime divisors of every a b a b with i=j. In other words,
i i j j i
− 6
we assume that
k
(2k)! a (a b a b ) is a Σ-unit. (3.5)
i i j j i
−
i=1 1≤i<j≤k
Y Y
Then there exist real numbers η,c (0,1/2], depending only on k and on the
1
∈
cardinality7 #Σ (but not on Σ or on the integers a and b ), and z 2 depend-
i i 1
≥
ing on a ,...,a ,b ,...,b , such that the following holds. For any x and z,
1 k 1 k
satisfying xη z z , we have #✵ (x,z) c x(logz)−k.
1 1 1
≥ ≥ ≥
7Indicating dependence onk hereisabitobsolete, because our hypothesis impliesthatk
isboundedintermsof#Σ.
6
InSubsection3.2,webrieflyrecallthebasicsofBrunsieve,followingmainly
theHalberstam-Richertclassicaltreatise[7]. InSubsection3.3,weusetheBrun
sievetoestablishtheorderofmagnitudeof#✵(x,z). InSubsections3.4and3.5,
weboundfromabovethenumberofunwantedn ✵(x,z),thoseforwhichsome
∈
L (n) is prime or non-squarefree. Finally, in Subsection 3.6, we complete the
i
proof of Theorem 3.1.
3.2 Outline of the Brun Sieve
In this subsection, we give a quick overview of the combinatorial Brun sieve
in the form used in sequel. Our principal reference is [7, Chapters 1 and 2],
especially Section 2.4. For the reader’s convenience, we try to follow the set-up
adopted in [7], with a few exceptions that will be highlighted below.
Recall that, in this article, d (with or without indices) stands for a square-
free positive integer. Foreverym Z, we associateanon-negativerealnumber
∈
a(m), such that a(m)=0 for all but finitely many m. For every d, we set
= a(m).
d
A
Xd|m
Remark 3.2. In [7], is a finite multi-set of integers (and is the subset
d
A A
of consistingoftheintegers divisible byd). Definingsuchamulti-setis equiv-
A
alent to defining a sequence a(m) as above consisting of non-negative integers.
Hencetheresultsfrom[7]areformally provedonlyforsuchsequences. However,
integrality of the numbers a(m) is never used in the arguments from [7] that we
quote, so the results actually hold true for any sequences of non-negative real
numbers.
We also fix a multiplicative function ρ (in [7] it is denoted by ω) supported
at square-free numbers and having the following properties.
A. There exists a real number A 1 such that for every prime p, we have
1
≥
ρ(p) 1
0 1 (3.6)
≤ p ≤ − A
1
(condition (Ω ) on page 29 of [7]).
1
B. Thereexistrealnumbersα>0(in[7]itisdenotedbyκ)andA 1such
2
≥
that for z w 2, we have
≥ ≥
ρ(p)logp z
αlog +A (3.7)
2
p ≤ w
w≤p<z
X
(condition (Ω (α)) on page 52 of [7]).
2
Next we fix a real number X >1 and write, for every d,
ρ(d)
= X +R ,
d d
A d
7
where R are some real numbers.
d
Let Σ be a set of prime numbers (it corresponds to the set P¯ in [7]); for
z 2, we set
≥
(z)= a(m), (3.8)
S
(m,PXΣ(z))=1
ρ(p)
W(z)= 1 , (3.9)
− p
p|PYΣ(z)(cid:18) (cid:19)
where P (z) is defined in (3.1).
Σ
The following statement is a version of Theorem 2.1 on page 57 of [7] (we
setthe parameterb therein to be 1). We denote by ω(d) the number ofdistinct
prime divisors of d.
Theorem 3.3. Let λ be a positive real number satisfying 0<λe1+λ <1. Set
A A 1 2λ 1
2 2
c = 1+A α+ , ε= , Λ= , (3.10)
1 2 1 log2 200e1/α α 1+ε
(cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19)
whereA ,A andαarefrom (3.6), (3.7). Then for z 2, wehave thefollowing
1 2
≥
estimates:
λ3e2λ 5c
1
(z) XW(z) 1+2 exp (3.11)
S ≤ 1 λ2e2+2λ λlogz
(cid:18) − (cid:18) (cid:19)(cid:19)
+ R , (3.12)
d
| |
d|PΣ(z),Xd<z2eΛΛ−1,
ω(d)<2Λ−1(loglogz−loglog2)+2
λ2e2λ 4c
1
(z) XW(z) 1 2 exp (3.13)
S ≥ − 1 λ2e2+2λ λlogz
(cid:18) − (cid:18) (cid:19)(cid:19)
R . (3.14)
d
− | |
d|PΣ(z), dX<z2eΛΛ−1−1,
ω(d)<2Λ−1(loglogz−loglog2)+1
The first summands in (3.11) and(3.13) (those of the form XW(z)( ) are
···
exactly the same as in equations (4.2) and (4.3) on page 57 of [7], if you set
b=1 therein. However, our “error term” (the sum involving R ) is totally
d
| |
different. The reasonis that in [7] condition R ρ(d) (that is, condition (R)
d
| |≤
on page 30) is imposed, and we do not want to impose it.
We briefly explain below how the argument on pages 57–62 of [7] can be
adapted to prove Theorem 3.3.
Proof of Theorem 3.3. Throughout the proof, we denote P (z) by P(z).
Σ
Let r be a positive integer and z ,...,z be real numbers satisfying
0 r
z =z >z >z > >z =2;
0 1 2 r
···
they will be specified later. We consider functions χ and χ defined on the set
1 2
of divisors of P(z) as follows:
8
χ (d)=1ifdhasatmost2nprimedivisorsintheinterval[z ,z)forevery
1 n
•
n=1,...,r, and χ (d)=0 otherwise;
1
χ (d)=1 if d has at most 2n 1 prime divisors in the interval [z ,z) for
2 n
• −
every n=1,...,r, and χ (d)=0 otherwise;
1
(These definitions are equivalent to equation (4.8) on page 58 of [7], if we set
b=1 therein.)
It is explained in [7] that with this definition of χ and χ we have
1 2
ρ(d)
(z) X µ(d)χ (d) + χ (d)R , (3.15)
1 1 d
S ≤ d | |
d|XP(z) d|XP(z)
ρ(d)
(z) X µ(d)χ (d) χ (d)R . (3.16)
2 2 d
S ≥ d − | |
d|XP(z) d|XP(z)
See equation (4.5) on page 57 of [7].
Now let us specify z ,...,z . We follow [7, page 60]. Choose r to be the
0 r
only positive integer with the property
logz
e(r−1)Λ < erΛ, (3.17)
log2 ≤
where Λ is defined in (3.10). Now define the numbers z by
n
logz =e−nΛlogz (n=0,1,...,r 1), z =2.
n r
−
Itisshownonpages57–62of[7]thatwiththischoiceofz wehavetheestimates
n
ρ(d) λ3e2λ 5c
1
µ(d)χ (d) W(z) 1+2 exp ,
1 d ≤ 1 λ2e2+2λ λlogz
d|XP(z) (cid:18) − (cid:18) (cid:19)(cid:19)
ρ(d) λ2e2λ 4c
1
µ(d)χ (d) W(z) 1 2 exp .
2 d ≥ − 1 λ2e2+2λ λlogz
d|XP(z) (cid:18) − (cid:18) (cid:19)(cid:19)
So, we are left with the error terms.
Our definition of r implies that
r <Λ−1(loglogz loglog2)+1.
−
Hence, for d with χ (d)=1, we have
1
ω(d)<2r 2Λ−1(loglogz loglog2)+2.
≤ −
Furthermore, it follows from our definitions that for such d we have
d z2 z2 z2(1+e−Λ+···+e−(r−1)Λ) <z2(1−e−Λ)−1 <z2eΛΛ−1.
≤ 0··· r−1 ≤
9
Hence,
χ (d)R R .
1 d d
| |≤ | |
d|XP(z) d|P(z), dX<z2eΛΛ−1,
ω(d)<2Λ−1(loglogz−loglog2)+2
In a similar way, one proves that
χ (d)R R .
2 d d
| |≤ | |
d|XP(z) d|P(z), d<Xz2eΛΛ−1−1,
ω(d)<2Λ−1(loglogz−loglog2)+1
This completes the proof of Theorem 3.3.
Here is a much less precise, but more compact and convenient version of
Theorem 3.3.
Corollary 3.4. In the set-up of Theorem 3.3 assume that α 1. Then there
≥
exists z 2, depending on α, A and A , such that for z z , we have
0 1 2 0
≥ ≥
1
XW(z) (z) (z) 2XW(z)+ (z),
2 −R ≤S ≤ R
where
(z)= R .
d
R | |
d|PΣ(zX), d<z20α,
ω(d)<20αloglogz
Proof. Set in Theorem 3.3 λ=0.1. Then 0.2 Λ (6α)−1, and the result
≥ ≥
follows by an easy calculation.
3.3 Sieving away small primes factors
In this and the subsequent subsections we use the notation and the set-up of
Subsection 3.1.
Proposition3.5. Intheset-upofTheorem3.1,thereexistsrealnumbersz 2
0
≥
and c (0,1/2], both depending only on k such that for x and z, satisfying
∈
x z50k and z z we have
0
≥ ≥
x x
c #✵(x,z) c−1 2#Σ . (3.18)
(logz)k ≤ ≤ · (logz)k
Proof. In this proof, unless the contrary is stated explicitly, the constants im-
plied by the notation O(), , or8 , may depend only on k. The same
· ≪ ≫ ≍
convention applies to the the constants implied by the expressions like “suffi-
ciently large”.
Letρbethemultiplicativefunctionconcentratedonthesquare-freenumbers
and such that
k, p / Σ,
ρ(p)= ∈
(0, p Σ.
∈
8WeuseA≍B asashortcutforA≪B≪A.
10