AN ANALOGUE OF THE ERDÖS-KAK THEOREM FOR
FOURIER COEFFICIENTS OF MODULAR FORMS
M. RAM MURTY AND V. KUMAR MURTY
Abstract. Let f be a cusp form of weight k on Γ0 (N ) which is a normalized
eigenform for the Hecke operators,
and suppose that f does not have complex
P 2πinz
multiplication. Write f =
an e
for the Fourier expansion at ∞ and
suppose an ∈ Z for all n. Denote by ν(n) the number of distinct prime divisors
of n. Assuming the Riemann Hypothesis for all Artin L-functions, we show
that
card p ≤ X : ap 6= 0 and ν(ap ) − log log p ≤ α(log log p)1/2

1
∼ √
2π
Z∞
e
−t2 /2
−∞

dt π(X).
An analogous result is obtained for Ω(ap ), the total number of prime factors
of ap . We also obtain the distribution function for ν(an ) and Ω(an ).
1. Introduction
Let f be a normalized eigenform of the Hecke operators for Γ0 (N ). Suppose that
f does not have complex multiplication (in the sense of Ribet) and that
f (z) =
∞
X
an e2πinz
n=1
with an ∈ Z. It was shown in Murty and Murty (1984) that, assuming a certain
generalized Riemann hypothesis (GRH),
X
(ν(ap ) − log log p)2 ≪
p≤x
ap 6=0
x log log x
log x
(1)
and
X
1
(ν(ap ) − (log log n)2 )2 ≪ x(log log x)3+ε
2
(2)
p≤x
ap 6=0
where ν(n) denotes the number of prime factors of n and in (1), p denotes a prime
number.
These results led to the interesting conclusion that
|ap | ≥ exp((log p)1−ε )
1
(3)
2
M. RAM MURTY AND V. KUMAR MURTY
for almost all primes p (that is, apart from o(x/ log x) primes p ≤ x, the inequality
(3) holds.) It is a classical conjecture of Lehmer, that in the case f = ∆, the
Ramanujan cusp form,
τ (p) 6= 0.
Here τ denotes the Ramanujan function. Serre and Atkin have further conjectured
that
|τ (p)| ≥ p9/2−ε
for all p sufficiently large. If we write τ (p) = 2p11/2 cos θp and assume the SatoTate conjecture, namely that θp ’s are equidistributed in [−π, π] with respect to the
measure (2/π) sin2 θ dθ, then it is easy to see that
|τ (p)| ≥ p11/2−ε
for almost all prime numbers p.
(1) and (2) can be viewed as the ”modular” analogue of the classical theorem of
Hardy and Ramanujan concerning the function ν(n). It was also noted in Murty
and Murty (1984) that the assumption of GRH in (1) and (2) can be replaced by
a (milder) inequality of Bombieri-Vinogradov type. The exact formulation will be
given below.
The purpose of this paper is to prove that ν(τ (n)) and ν(τ (p)) obey a normal
distribution law very much analogous to ν(n) (as described by the celebrated ErdösKak theorem).
More generally, let g be a non-zero multiplicative function taking rational integer
values. Suppose that g(n) 6= 0 for any natural number n, and
Hypothesis 1. For some β > 0, |g(n)| ≤ nβ for all n.
Define,
π(x; g, d) = card(p ≤ x : g(p) ≡ 0 (mod d))
where (here and elsewhere in this paper) p denotes a prime number.
We suppose that there is a function δ(d) such that:
Hypothesis 2. For some θ > 0,
X
|π(x; g, d) − δ(d)π(x)| ≪
d≤xθ
x
log x
where π(x) denotes the number of primes p ≤ x.
We make the following hypothesis purely for the sake of clarity of exposition.
More general results can be proved under less restrictive conditions. Nevertheless,
we assume:
Hypothesis 3. For prime powers pα , q β , (p 6= q),
δ(pα ) = p−α + O(p−α−1 )
and
δ(pα q β ) = (p−α + O(p−α−1 ))(q −β + O(q −β−1 ))
where the implied constants are absolute.
Define:
ν(g(n)) − 12 (log log n)2
α
√
≤
N (x, α) = card n ≥ x :
.
(log log n)3/2
3
We can now state the main theorem:
AN ANALOGUE OF THE ERDÖS-KAK THEOREM FOR FOURIER COEFFICIENTS OF MODULAR FORMS
3
Theorem 1. Let g be a non-vanishing multiplicative function satisfying Hypotheses
1, 2, and 3 above. Then
Z∞
2
N (x, α)
1
lim
e−t /2 dt.
= √
x→∞
x
2π
−∞
Now let f be a Hecke eigenform on Γ0 (N ), with rational integer coefficients an .
We modify the coefficients by defining:
Y
apα .
af
n =
pα kn
apα 6=0
The empty product is taken to be equal to 1.
Define
!
1
2
g
v (a
α
n ) − 2 (log log n)
.
Nf (x, α) = card n ≤ x :
≤ √
(log log n)3/2
3
Theorem 2. Let f be a normalized Hecke eigenform on Γ0 (N ) with rational integer
coefficients and without complex multiplication. Then
Z∞
2
Nf (x, α)
1
lim
= √
e−t /2 dt.
x→∞
x
2π
−∞
2. The Theorem of Kubilius-Shapiro
An additive arithmetical function h is called strongly additive if h(pα ) = h(p).
There is a general theorem of Kubilius and Shapiro which allows one to deduce a
normal distribution law for such functions satisfying certain conditions. We would
like to apply this theorem to ν(g(n)), but a little thought reveals that ν(g(n)) is
not additive. Our basic idea is to approximate this function by a strongly additive function and apply the general theory. Finally, we show that the error thus
introduced is small enough to deduce our main results.
Given any strongly additive function h, define
X h(p)
A(x) =
p
p≤x
and
1/2
X h2 (p)
 .
B(x) = 
p

Suppose that for each fixed ε > 0,
X
p≤x
|h(p)|>εB(x)
Then, setting
p≤x
h2 (p)
= o(B 2 (x)).
p
h(n) − A(x)
≤α
H(x; α) = card n ≤ x :
B(x)
4
M. RAM MURTY AND V. KUMAR MURTY
the theorem of Kubilius-Shapiro (see Elliott 1980b) implies
1
H(x; α)
= √
lim
x→∞
x
2π
Z∞
2
e−t
/2
dt.
−∞
3. The First Approximation
Let Ω(n) denote the total number of prime factors of n, counted with multiplicity.
(Note that Ω is ‘not’ strongly additive.) Define
X
Ω(g(p)).
h(n) =
p|n
Clearly h is strongly additive. We have
X h(p)
X Ω(g(p))
=
p
p
p≤x
p≤x
X
=
qα ≤xθ
X 1
+ O(log log x)
p
p≤x
qα | g(p)
since the number of prime power divisors q α of g(p) with q α > xθ is absolutely
bounced by virtue of Hypothesis 1. Partial summation gives that the first sum is


Z∞
 X
X 1
X
dt
π(x; g, q α ) + π(t; g, q α ) 2 =
+
(say).
1
2
x
t 
α
θ
q ≤x
2
Now, by Hypothesis 2 and Hypothesis 3, we have
X
log log x
.
≪
1
log x
To handle the other sum, we need the following lemmas:
Lemma 1. If u ≤ xθ , then
X
π(x; g, d) = π(x) log log u + O(π(x)).
d≤u
If u > xθ , then
X
π(x; g, d) = π(x) log log x + O(π(x)).
d≤u
Proof. This is easily established utilising Hypothesis 1 and Hypothesis 2.
By lemma 1, we fine
X
Therefore,
2
Z∞
dt
= {π(t) log log t + O(t/ log t)} 2 .
t
2
X h(p)
1
= (log log x)2 + O(log log x).
p
2
p≤x
AN ANALOGUE OF THE ERDÖS-KAK THEOREM FOR FOURIER COEFFICIENTS OF MODULAR FORMS
5
Now we treat B(x) for the function h. We find,
X1 X
B 2 (x) =
1 + O(x(log log x)2 ).
p α
p≤x
q1 | g(p)
q2γ | g(p)
q1 6=q2
Again by partial summation,
B 2 (x)
=


∞
X  π(x; g, q α q γ ) Z
dt 
1 2
+ π(t; g, q1α q2γ ) 2

x
t 
α
θ/2
q1 ≤x
q2γ ≤xθ/2
=
Z∞ X
2
δ(q1α q2γ )π(t)
α
θ/2
2 q1 ≤t
q2γ ≤tθ/2
dt
+ O((log log x)2 )
t2
by Lemma 1 and Hypothesis 2. We find easily that
Z
(log log t)2
dt + O((log log x)2 )
B 2 (x) = x
t log t
2
utilising Hypothesis 3. Therefore,
1
B(x) = √ (log log x)3/2 + O(log log x).
3
We need one more calculation. We have to check that:
X
Ω2 (g(p))
= o(B 2 (x)).
p
(4)
p≤x
Ω(g(p))>εB(x)
Define
α(p) =
(
1
0
if Ω(g(p)) > εB(x)
if not.
It is not difficult to see (by Turan’s method and utilising Hypotheses 1, 2, and 3)
that
X
(Ω(g(p)) − log log p)2 = O(x log log x).
p≤x
This was in fact established in Murty and Murty (1984).
Therefore,
X
α(p) = O(x/(log log x)2 ).
p≤x
Hence
2
X
Ω (g(p)) α(p)
·
p
p≤x
2 
1/2
X α(p)
X Ω4 (g(p))
 


p
p
≤

≪

utilising (5) and partial summation.
p≤x

p≤x
1/2
X Ω4 (g(p))

p
p≤x
(5)
6
M. RAM MURTY AND V. KUMAR MURTY
We readily find under Hypotheses 1, 2, and 3, the estimate
X Ω4 (g(p))
≪ (log log x)5 .
p
p≤x
In fact, one can establish an asymptotic formula by the preceding method, but it
is not necessary for our purpose. We therefore deduce that (4) does indeed hold.
This proves that


Z∞
h(n) − 21 (log log n)2
2
1
α
= √
e−t /2 dt + o(1) x.
card n ≤ x :
≤√
(log log n)3/2
3
2π
−∞
4. The Turan-Kubilius Inequality
We want information concerning ν(g(n)), and Ω(g(n)). The first part of Theorem
1, for N (x, α) will be established if we can show that
ν(g(n)) − h(n) = o((log log x)3/2 )
for all but o(x) numbers n ≤ x. For this purpose, we need:
Lemma 2. (Turan-Kubilius inequality) — Let k(n) be a complex-valued additive function defined on the natural numbers. If
X k(pα ) 1
1
−
E(x) =
pα
p
α
p ≤x
and
1/2
X |k(pα )|

D(x) = 
pα
α

then
X
p ≤x
|k(n) − E(x)|2 ≤ 32xD2 (x).
n≤x
Proof. See Elliott (1980a, p. 147).
First, we establish that for almost all n,
Ω(g(n)) − Ωy (g(n)) = ν(g(n)) − νy (g(n))
where νy (n) and Ωy (n) are the number of prime factors of n lesss than y counted
without multiplicity and with multiplicity respectively.
The function Ω(g(n)) is additive. Setting
X Ω(g(pα )) 1
1−
E(x) =
pα
p
α
p ≤x
=
X Ω(g(p))
+ O(1)
p
p≤x
and D(x) = B(x), Lemma 2 gives
2
X
1
Ω(g(n)) − (log log n)2
≪ x(log log x)3 .
2
n≤x
AN ANALOGUE OF THE ERDÖS-KAK THEOREM FOR FOURIER COEFFICIENTS OF MODULAR FORMS
7
If we consider Ωy (g(n)) with y = log log x, it is easy to see that the corresponding
E(x) and D(x), which we shall denote by Ey (x) and Dy (x), satisfy
Ey (x) = (log2 x)(log4 x) + O(log2 x)
and
Dy (x)2 = (log2 x)(log4 x)2 + O((log2 x)(log4 x))
where we have used the notation of the iterated logarithm: log1 x = log x, logk x =
log(logk−1 x), for k ≥ 2. Again, by Lemma 2,
X
2
(Ωy (g(n)) − (log2 x)(log4 x)) ≪ x(log2 x)(log4 x)2 .
n≤x
Therefore,
Ωy (g(n)) < 2(log2 x)(log4 x)
(6)
for almost all n ≤ x. Hence
0 ≤ Ωy (g(n)) − νy (g(n)) ≤ 2(log2 x)(log4 x)
for almost all n ≤ x.
Now suppose p > y and p | g(n). Then p | g(q α ) for some prime power q α k n.
The quantity
Ω(g(n)) − Ωy (g(n))
counts prime divisors p of g(n), p > y with multiplicity, and
ν(g(n)) − νy (g(n))
counts the same without multiplicity. Suppose p2 | g(n) and p > y. Then there are
two possibilities:
(a) there is a q α k n such that g(q α ) ≡ 0 (mod p2 )
or
(b) there are q1α , q2γ k n wuch that g(q1α ) ≡ 0 (mod p) and g(q2γ ) ≡ 0 (mod p).
In case (a), the number of n ≤ x is
X
X x
≪
= o(x)
qα
α
β 2
y<p<x p | g(q )
by partial summation, Hypothesis 1 and Hypothesis 2. In case (b), we get that the
number of such natural numbers is
X
X
x
≪
α q γ = o(x)
q
α
1 2
β
y<p<x p | g(q1 )
p | g(q2γ )
by a similar method. Therefore, for almost all n,
Ω(g(n)) − Ωy (g(n)) = ν(g(n)) − νy (g(n))
and the dispersion method of Lemma 2 yields (6) and therefore, if the theorem is
true for Ω(g(n)), it is certainly true for ν(g(n)).
To this end, we establish that
Ω(g(n)) − Ωy (g(n)) = h(n) − hy (n)
for almost all n ≤ x. Here,
hy (n) =
X
p|n
p<y
Ω(g(p)).
8
M. RAM MURTY AND V. KUMAR MURTY
We begin by noting that
Ω(g(n)) − h(n) =
X
α
p kn
α≥2
(Ω(g(pα )) − Ω(g(p))) .
As h(n) and Ω(g(n)) are additive functions, the dispersion Lemma 2 applies. If we
set
j(n) = Ω(g(n)) − h(n)
we find that if
X j(pα ) 1
1−
c=
pα
p
α
p
α≥2
then
X
(j(n) − c)2 ≪ x.
n≤x
Therefore, for almost all n,
Ω(g(n)) = h(n) + o((log log n)3/2 ).
5. Proofs of Theorems 1 and 2
By the above, we have established that
Z∞
Ω(g(n)) − 12 (log log n)2
2
α
1
√
√
e−t /2 dt
≤
Pr n :
=
(log log n)3/2
3
2π
−∞
and our previous remarks show that the same is true for ν(g(n)).
For Theorem 2, it was already noted in Murty and Murty (1984) that Fourier
coefficients of normalized Hecke eigenforms of Γ0 (N ) satisfy Hypotheses 1, 2 and 3
via the associated l-adic representation. We will not repeat this here. Theorem 2
now follows from Theorem 1 after this observation.
6. Concluding Remarks
If we let
ν(g(p)) − log log p
P (x, α) = card p ≤ x :
≤
α
(log log p)1/2
then utilising the methods of this paper, one can show that for each fixed k,
X
x
(log log x)k/2
(ν(g(p)) − log log p)k = (ck + o(1))
log x
p≤x
where
1
ck = √
2π
Z∞
2
tk e−t
/2
dt.
−∞
The case g(p) = p − 1 was treated by Halberstam (1955). It follows from Tchebycheff’s method of moments that
Z ∞
2
P (x, α)
1
e−t /2 dt.
lim
= √
x→∞ x/ log x
2π −∞
AN ANALOGUE OF THE ERDÖS-KAK THEOREM FOR FOURIER COEFFICIENTS OF MODULAR FORMS
9
This of course yields a corresponding result for the Fourier coefficients of normalized
eigenforms. In fact, since
card(p ≤ x : ap = 0) = o(x/ log x)
by a result of Serre (1982), we have:
Theorem 3. Lef f be a normalized Hecke eigenform on Γ0 (N ) with rational integer
coefficients and without complex multiplication. Then
ν(ap ) − log log p
√
≤ α = (c + o(1))π(x)
card p ≤ x : ap 6= 0 and
log log p
where
Z∞
2
1
c= √
e−t /2 dt.
2π
−∞
In the special case of g(n) = φ(n) in Theorem 1, we recover the theorem of
Erdös and Pomerance (1984). In fact, our method has been inspired by their work.
Hypothesis 2 is verified to hold by invoking the Bombieri-Vinogradov theorem.
Hypothesis 2 is actually too strong an assumption, as was already pointed out
in Murty and Murty (1984). A suitable analogue of the classical Rodosski and
Tatuzawa would suffice for our purposes. It would be a fruitful investigation to find
out if such an analogue exists for Artin L-series.
References
[1] Elliott, P.D. (1980a). Probabilistic Number Theory I, Mean Value Theorems. Springer
Verlag, Berlin.
[2] Elliott, P.D. (1980b). Probabilistic Number Theory II, Central Limit Theorems. SpringerVerlag, Berlin.
[3] Erdös, P., and Pomerance, C. (1984). On the normal number of prime factors of φ(n).
Proceedings of Edmonton Number Theory Conference, (to appear).
[4] Halberstam, H. (1955). On the distribution of additive number theoretic functions I. J.
Lond. Math. Soic., 30, 43–53.
[5] Ram Murty, M., Kumar Murty, V. (1984). Prime divisors of Fourier coefficients of modular
forms. Duke. Math. J., 51, 57–76.
[6] Serre, J-P. (1982). Quelques applications du théorème de densité de Chebotarev. Publ.
Math. I.H.E.S., 54, 123–201.
Institute for Advanced Study, Princeton, NJ 08540, USA
Tata Institute of Fundamental Research, Bombay 400005