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 Γ_{0}(N) which is a normalized eigenform for the Hecke operators, and suppose that f does not have complex multiplication. Write f = ∑ a_{n}e^{2πinz} for the Fourier expansion at ∞ and suppose a_{n} ∈ 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 : a_{p} ≠ 0 and ν(a_{p}) − log log p ≤ α(log log p)^{1/2}}

∼ (\frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−t^{2}/2} dt) π(X).

An analogous result is obtained for Ω(a_{p}), the total number of prime factors of a_{p}. We also obtain the distribution function for ν(a_{n}) and Ω(a_{n}).

1. Introduction

Let f be a normalized eigenform of the Hecke operators for Γ_{0}(N). Suppose that f does not have complex multiplication (in the sense of Ribet) and that

f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz}

with a_{n} ∈ ℤ. It was shown in Murty and Murty (1984) that, assuming a certain generalized Riemann hypothesis (GRH),

∑_{\begin{matrix}p≤x \\ a_{p}≠0 \end{matrix}}(ν(a_{p}) − log log p)^{2} ≪ \frac{x log log x}{log x}  (1)

and

∑_{\begin{matrix}p≤x \\ a_{p}≠0 \end{matrix}}(ν(a_{p}) − \frac{1}{2}(log log n)^{2})^{2} ≪ x(log log x)^{3+ε}  (2)

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

|a_{p}| ≥ exp((log p)^{1−ε})  (3)

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 = Δ, the Ramanujan cusp form,

τ(p) ≠ 0.

Here τ denotes the Ramanujan function. Serre and Atkin have further conjectured that

|τ(p)| ≥ p^{9/2−ε}

for all p sufficiently large. If we write τ(p) = 2p^{11/2} cos θ_{p} and assume the Sato-Tate conjecture, namely that θ_{p}’s are equidistributed in [−π, π] with respect to the measure (2/π) sin^{2} θ dθ, then it is easy to see that

|τ(p)| ≥ p^{11/2−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 Erdös-Kak theorem).

More generally, let g be a non-zero multiplicative function taking rational integer values. Suppose that g(n) ≠ 0 for any natural number n, and

Hypothesis 1. For some β > 0, |g(n)| ≤ n^{β} 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,

∑_{d≤x^{θ}}|π(x; g, d) − δ(d)π(x)| ≪ \frac{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 p^{α}, q^{β}, (p ≠ q),

δ(p^{α}) = p^{−α} + O(p^{−α−1})

and

δ(p^{α}q^{β}) = (p^{−α} + O(p^{−α−1}))(q^{−β} + O(q^{−β−1}))

where the implied constants are absolute.

Define:

N(x, α) = card (n ≥ x : \frac{ν(g(n)) − \frac{1}{2}(log log n)^{2}}{(log log n)^{3/2}} ≤ \frac{α}{\sqrt{3}}) .

We can now state the main theorem:

Theorem 1. Let g be a non-vanishing multiplicative function satisfying Hypotheses 1, 2, and 3 above. Then

lim_{x→∞} \frac{N(x, α)}{x} = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−t^{2}/2} dt.

Now let f be a Hecke eigenform on Γ_{0}(N), with rational integer coefficients a_{n}. We modify the coefficients by defining:

ã_{n} = ∏_{\begin{matrix}p^{α}∥n \\ a_{p^{α}} ≠0 \end{matrix}} a_{p^{α}}.

The empty product is taken to be equal to 1.

Define

N_{f}(x, α) = card (n ≤ x : \frac{v ˜(a_{n}) − \frac{1}{2}(log log n)^{2}}{(log log n)^{3/2}} ≤ \frac{α}{\sqrt{3}}) .

Theorem 2. Let f be a normalized Hecke eigenform on Γ_{0}(N) with rational integer coefficients and without complex multiplication. Then

lim_{x→∞} \frac{N_{f}(x, α)}{x} = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−t^{2}/2} dt.

2. The Theorem of Kubilius-Shapiro

An additive arithmetical function h is called strongly additive if h(p^{α}) = 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

A(x) = ∑_{p≤x} \frac{h(p)}{p}

and

B(x) = (∑_{p≤x} \frac{h^{2}(p)}{p})^{1/2} .

Suppose that for each fixed ε > 0,

∑_{\begin{matrix}p≤x \\ |h(p)|>εB(x) \end{matrix}} \frac{h^{2}(p)}{p} = o(B^{2}(x)).

Then, setting

H(x; α) = card (n ≤ x : \frac{h(n) − A(x)}{B(x)} ≤ α)

the theorem of Kubilius-Shapiro (see Elliott 1980b) implies

lim_{x→∞} \frac{H(x; α)}{x} = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−t^{2}/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

h(n) = ∑_{p | n} Ω(g(p)).

Clearly h is strongly additive. We have

∑_{p≤x} \frac{h(p)}{p} = ∑_{p≤x} \frac{Ω(g(p))}{p}

= ∑_{q^{α}≤x^{θ}} ∑_{\begin{matrix}p≤x \\ q^{α} | g(p) \end{matrix}} \frac{1}{p} + O(log log x)

since the number of prime power divisors q^{α} of g(p) with q^{α} > x^{θ} is absolutely bounced by virtue of Hypothesis 1. Partial summation gives that the first sum is

∑_{q^{α}≤x^{θ}} \{\frac{1}{x}π(x; g, q^{α}) + ∫_{2}^{∞}π(t; g, q^{α})\frac{dt}{t^{2}} \} = _{∑1} +_{∑2} (say).

Now, by Hypothesis 2 and Hypothesis 3, we have

∑1 ≪ \frac{log log x}{log x}.

To handle the other sum, we need the following lemmas:

Lemma 1. If u ≤ x^{θ}, then

∑_{d≤u} π(x; g, d) = π(x) log log u + O(π(x)).

If u > x^{θ}, then

∑_{d≤u} π(x; g, d) = π(x) log log x + O(π(x)).

Proof. This is easily established utilising Hypothesis 1 and Hypothesis 2.

By lemma 1, we fine

∑2 = ∫_{2}^{∞}\{π(t) log log t + O(t/ log t)\} \frac{dt}{t^{2}}.

Therefore,

∑_{p≤x} \frac{h(p)}{p} = \frac{1}{2}(log log x)^{2} + O(log log x).

Now we treat B(x) for the function h. We find,

B^{2}(x) = ∑_{p≤x} \frac{1}{p} ∑_{\begin{matrix}q_{1}^{α} | g(p) \\ q_{2}^{γ} | g(p) \\ q_{1}≠q_{2} \end{matrix}} 1 + O(x(log log x)^{2}).

Again by partial summation,

B^{2}(x) = ∑_{\begin{matrix}q_{1}^{α}≤x^{θ/2} \\ q_{2}^{γ}≤x^{θ/2} \end{matrix}} \{\frac{π(x; g, q_{1}^{α}q_{2}^{γ})}{x} + ∫_{2}^{∞}π(t; g, q_{1}^{α}q_{2}^{γ}) \frac{dt}{t^{2}} \}

= ∫_{2}^{∞} ∑_{\begin{matrix}q_{1}^{α}≤t^{θ/2} \\ q_{2}^{γ}≤t^{θ/2} \end{matrix}} δ(q_{1}^{α}q_{2}^{γ})π(t) \frac{dt}{t^{2}} + O((log log x)^{2})

by Lemma 1 and Hypothesis 2. We find easily that

B^{2}(x) = ∫_{2} x\frac{(log log t)^{2}}{t log t} dt + O((log log x)^{2})

utilising Hypothesis 3. Therefore,

B(x) = \frac{1}{\sqrt{3}}(log log x)^{3/2} + O(log log x).

We need one more calculation. We have to check that:

∑_{\begin{matrix}p≤x \\ Ω(g(p))>εB(x) \end{matrix}} \frac{Ω^{2}(g(p))}{p} = o(B^{2}(x)).  (4)

Define

α(p) = \begin{cases}1 & if Ω(g(p)) > εB(x) \\ 0 & if not. \end{cases}

It is not difficult to see (by Turan’s method and utilising Hypotheses 1, 2, and 3) that

∑_{p≤x}(Ω(g(p)) − log log p)^{2} = O(x log log x).

This was in fact established in Murty and Murty (1984).

Therefore,

∑_{p≤x} α(p) = O(x/(log log x)^{2}).  (5)

Hence

|∑_{p≤x} α(p) · \frac{Ω^{2}(g(p))}{p}| ≤ (∑_{p≤x} \frac{α(p)}{p})^{2} (∑_{p≤x} \frac{Ω^{4}(g(p))}{p})^{1/2}

≪ (∑_{p≤x} \frac{Ω^{4}(g(p))}{p})^{1/2}

utilising (5) and partial summation.

We readily find under Hypotheses 1, 2, and 3, the estimate

∑_{p≤x} \frac{Ω^{4}(g(p))}{p} ≪ (log log x)^{5}.

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

card (n ≤ x : \frac{h(n) − \frac{1}{2}(log log n)^{2}}{(log log n)^{3/2}} ≤ \frac{α}{\sqrt{3}}) = (\frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−t^{2}/2} dt + o(1)) x.

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

E(x) = ∑_{p^{α}≤x} \frac{k(p^{α})}{p^{α}} (1 − \frac{1}{p})

and

D(x) = ( ∑_{p^{α}≤x} \frac{|k(p^{α})|}{p^{α}} )^{1/2}

then

∑_{n≤x}|k(n) − E(x)|^{2} ≤ 32xD^{2}(x).

Proof. See Elliott (1980a, p. 147).

First, we establish that for almost all n,

Ω(g(n)) − Ω_{y}(g(n)) = ν(g(n)) − ν_{y}(g(n))

where ν_{y}(n) and Ω_{y}(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

E(x) = ∑_{p^{α}≤x} \frac{Ω(g(p^{α}))}{p^{α}} (1 − \frac{1}{p})

= ∑_{p≤x} \frac{Ω(g(p))}{p} + O(1)

and D(x) = B(x), Lemma 2 gives

∑_{n≤x} (Ω(g(n)) − \frac{1}{2}(log log n)^{2})^{2} ≪ x(log log x)^{3}.

If we consider Ω_{y}(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 E_{y}(x) and D_{y}(x), satisfy

E_{y}(x) = (log_{2} x)(log_{4} x) + O(log_{2} x)

and

D_{y}(x)^{2} = (log_{2} x)(log_{4} x)^{2} + O((log_{2} x)(log_{4} x))

where we have used the notation of the iterated logarithm: log_{1} x = log x, log_{k} x = log(log_{k−1} x), for k ≥ 2. Again, by Lemma 2,

∑_{n≤x} (Ω_{y}(g(n)) − (log_{2} x)(log_{4} x))^{2} ≪ x(log_{2} x)(log_{4} x)^{2}.

Therefore,

Ω_{y}(g(n)) < 2(log_{2} x)(log_{4} x)  (6)

for almost all n ≤ x. Hence

0 ≤ Ω_{y}(g(n)) − ν_{y}(g(n)) ≤ 2(log_{2} x)(log_{4} 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} ∥ n. The quantity

Ω(g(n)) − Ω_{y}(g(n))

counts prime divisors p of g(n), p > y with multiplicity, and

ν(g(n)) − ν_{y}(g(n))

counts the same without multiplicity. Suppose p^{2} | g(n) and p > y. Then there are two possibilities:

(a) there is a q^{α} ∥ n such that g(q^{α}) ≡ 0 (mod p^{2}) or

(b) there are q_{1}^{α}, q_{2}^{γ} ∥ n wuch that g(q_{1}^{α}) ≡ 0 (mod p) and g(q_{2}^{γ}) ≡ 0 (mod p). In case (a), the number of n ≤ x is

≪ ∑_{y<p<x^{β}} ∑_{p^{2} | g(q^{α})} \frac{x}{q^{α}} = o(x)

by partial summation, Hypothesis 1 and Hypothesis 2. In case (b), we get that the number of such natural numbers is

≪ ∑_{y<p<x^{β}} ∑_{\begin{matrix}p | g(q_{1}^{α}) \\ p | g(q_{2}^{γ}) \end{matrix}} \frac{x}{q_{1}^{α}q_{2}^{γ}} = o(x)

by a similar method. Therefore, for almost all n,

Ω(g(n)) − Ω_{y}(g(n)) = ν(g(n)) − ν_{y}(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)) − Ω_{y}(g(n)) = h(n) − h_{y}(n)

for almost all n ≤ x. Here,

h_{y}(n) = ∑_{\begin{matrix}p | n \\ p<y \end{matrix}} Ω(g(p)).

We begin by noting that

Ω(g(n)) − h(n) = ∑_{\begin{matrix}p^{α} ∥ n \\ α≥2 \end{matrix}} (Ω(g(p^{α})) − 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

c = ∑_{\begin{matrix}p^{α} \\ α≥2 \end{matrix}} \frac{j(p^{α})}{p^{α}} (1 − \frac{1}{p})

then

∑_{n≤x}(j(n) − c)^{2} ≪ 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

Pr (n : \frac{Ω(g(n)) − \frac{1}{2}(log log n)^{2}}{(log log n)^{3/2}} ≤ \frac{α}{\sqrt{3}}) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−t^{2}/2} dt

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 Γ_{0}(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

P (x, α) = card (p ≤ x : \frac{ν(g(p)) − log log p}{(log log p)^{1/2}} ≤ α)

then utilising the methods of this paper, one can show that for each fixed k,

∑_{p≤x}(ν(g(p)) − log log p)^{k} = (c_{k} + o(1))\frac{x}{log x}(log log x)^{k/2}

where

c_{k} = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} t^{k}e^{−t^{2}/2} dt.

The case g(p) = p − 1 was treated by Halberstam (1955). It follows from Tchebycheff’s method of moments that

lim_{x→∞} \frac{P (x, α)}{x/ log x} = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−t^{2}/2} dt.

This of course yields a corresponding result for the Fourier coefficients of normalized eigenforms. In fact, since

card(p ≤ x : a_{p} = 0) = o(x/ log x)

by a result of Serre (1982), we have:

Theorem 3. Lef f be a normalized Hecke eigenform on Γ_{0}(N) with rational integer coefficients and without complex multiplication. Then

card (p ≤ x : a_{p} ≠ 0 and \frac{ν(a_{p}) − log log p}{\sqrt{log log p}} ≤ α) = (c + o(1))π(x)

where

c = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−t^{2}/2} dt.

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. Springer-Verlag, 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.

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