SOME Ω-RESULTS FOR RAMANUJAN’S τ-FUNCTION

M. RAM MURTY

1. Introduction

I take great pleasure and pride in speaking to you, on this sacred land of India, about a very important arithmetical function, first discovered by the famous Indian mathematician Srinivasa Ramanujan. He had the foresight and the intuition to recognize the importance of modular forms in the theory of numbers. He investigated [9], in rather great detail, one modular form, now called Ramanujan’s cusp form, defined by

Δ(z) = q ∏_{n=1}^{∞}(1 − q^{n})^{24}, q = e^{2πiz},

This can be expanded in a power series in q and we have

Δ(z) = ∑_{n=1}^{∞} τ(n)q^{n}

and τ is called Ramanujan’s τ-function. We have

τ(1) = 1, τ(2) = −24, τ(3) = 252, τ(4) = −1472, τ(5) = 4830.

Ramanujan made two conjectures:

(i) τ is a multiplicative function, (ii) for each prime p, |τ(p)| ≤ 2p^{11/2}. The first conjecture was proved by Mordell [7] in 1917 and generalized in a beautiful way by Hecke [5]. The second conjecture was generalized by Petersson [8] to include other modular forms. The full Ramanujan-Petersson conjecture was settled by Deligne [2] in 1976.

It is known that

τ(p^{α}) = p^{11α/2} \frac{sin(α + 1)θ_{p}}{sin θ_{p}}

where θ_{p} ∈ (0, 2π), by the result of Deligne. As τ is multiplicative, it follows that

τ(n) = O (n^{11/2} exp (\frac{c log n}{log log n})) .

It is conjectured that this bound is sharp. More precisely,

Conjecture .

τ(n) = Ω (n^{11/2} exp (\frac{c log n}{log log n})) .

for some c > 0.

There is a conjecture of Sato-Tate that the angles θ_{p} are equidistributed, as p varies, with respect to the measure

\frac{2}{π} sin^{2} θ dθ,

This is an unproved conjecture.

Even the weaker conjecture: there exists a φ < \frac{π}{3} and a δ > 0 such that card (p ≤ x : 0 ≤ θ_{p} ≤ φ) ≫ x^{δ}, is unknown. We shall prove below:

Theorem 1. (modulo weak Sato-Tate conjecture)

τ(n) = Ω (n^{11/2} exp (\frac{c log n}{log log n})) .

With respect to unconditional results, Rankin [10] showed

lim sup_{n→∞} \frac{|τ(n)|}{n^{11/2}} = +∞

and Joris [6] improved this to

τ(n) = Ω (n^{11/2} exp (c (log n)^{\frac{1}{22} −ε})) .

R. Balasubramanian and I [1], succeeded in showing:

Theorem 2.

τ(n) = Ω (n^{11/2} exp (c (log n)^{\frac{2}{3} −ε})) .

2. Preliminaries

First, let me prove that τ(n) = O(n^{6}), so that the audience may gain some familiarity with the ideas involved in the latter part of this talk. The cusp form Δ(z) enjoys some properties, namely,

Δ (\frac{az + b}{cz + d}) = (cz + d)^{12}Δ(z)

for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ), and

SL_{2}(ℤ) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) : ad − bc = 1, a, b, c, d ∈ ℤ\} .

SL_{2}(ℤ) acts on H, the upper half plane. The standard fundamental domain for the action of SL_{2}(ℤ) on H is −\frac{1}{2} ≤ x ≤ \frac{1}{2} and x^{2} +y^{2} ≥ 1, where z = x + iy. The map-

ping z ↦ −\frac{1}{z} corresponding to ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) transforms the standard fundamental

domain to another fundamental domain.

Now y^{6} |Δ(z)| is easily checked to be invariant under the action of SL_{2}(ℤ) and

∫_{0}^{1} Δ(x + iy)e^{−2πimx} dx = τ(m)e^{−2πmy},

we find

y^{6}τ(m)e^{−2πmy} = O(1).

Choosing y = \frac{1}{m}, gives τ(m) = O(m^{6}).

For the sake of brevity, let us write

τ_{n} = τ(n)/n^{11/2}

and set

f(s) = ∑_{1}^{∞} \frac{τ_{n}^{2}}{n^{s}} .

Proof of Theorem 1. Now suppose the Sato-Tate conjecture is true. Then card (p ≤ x : 0 ≤ θ_{p} ≤ \frac{π}{6} ) is ∼ \frac{cx}{log x}. Take p ≤ x, such that 0 ≤ θ_{p} ≤ \frac{π}{6} . Then if N denotes the product of all such primes ≤ x, we have

log N ∼ cx

and so

τ_{N} ≥ (\sqrt{3})^{cx/ log x}
≥ exp (\frac{c log N}{log log N} ) ,

as desired.

The proof assuming the weak Sato-Tate conjecture is similar and left to the reader as an exercise.

3. Real Zeroes of f(s)

Let

ψ(s) = (2π)^{−2(s+11)}Γ(s + 11)Γ(s)ζ(2s)f(s)s(s − 1).

Then, it is known [4] that

ψ(s) = ∫∫_{D} y^{12} |Δ(z)|^{2} ϕ(z, s)\frac{dxdy}{y^{2}} ,

where D denotes the standard fundamental domain and

ϕ(z, s) = \frac{s(s − 1)}{z} (\frac{y}{π} )^{3} Γ(s)^{∑′} |mz + n|^{−2s} ,

the dash on the summation indicating that the sum is over all pairs of integers (m, n) ≠ (0, 0).

We shall sketch the proof that ψ(s) has no real zeroes. It is known that ϕ(z, s) > 0 for \frac{1}{2} ≤ s ≤ 1 and y ≤ 7, z = x + iy. Therefore, we split the fundamental domain into two regions: y ≤ A and y > A with A ≤ 7. In y ≤ A, ϕ(z, s) > 0 and the total contribution is > 0. In y > A, the contribution is small in virtue of the fact that Δ(z) = O(e^{−2πy}). The total integral is therefore > 0. A more precise proof requires the use of Kronecker’s limit formula, and we refer the reader to [1] for the details.

4. Proof of Theorem 2

We already saw that if we had τ_{p} > 1 for a large proportion of the primes, the Ω-result would follow. This leads us to consider τ_{p}^{2} − 1 and to study

∑_{p} \frac{τ_{p}^{2} − 1}{p^{s}}.

If we let

ϕ(s) = \frac{ζ(2s)}{ζ(s)}f(s),

we find

log ϕ(s) = ∑_{p} \frac{τ_{p}^{2}}{p^{s}} (1 + \frac{τ_{p}^{2} − 3}{2p^{s}} + ⋯) .

We need: Lemma 1.

∑_{\begin{matrix}p \\ τ_{p}^{2}>1 \end{matrix}} \frac{τ_{p}^{2} − 1}{p^{β}} = +∞ if β < \frac{1}{2}.

Proof. Suppose not. Then, for some β_{0} < \frac{1}{2},

∑_{τ_{p}^{2}>1} \frac{τ_{p}^{2} − 1}{p^{β_{0}}} < ∞

and so if we let

log ϕ(s) = f_{+}(s) − f_{−}(s),

where

f_{+}(s) = ∑_{\begin{matrix}p \\ τ_{p}^{2}>1 \end{matrix}} \frac{τ_{p}^{2} − 1}{p^{s}} (1 + \frac{τ_{p}^{2} − 3}{2p^{s}} + ⋯)

and

f_{−}(s) = − ∑_{\begin{matrix}p \\ τ_{p}^{2}<1 \end{matrix}} \frac{τ_{p}^{2} − 1}{p^{s}} (1 + \frac{τ_{p}^{2} − 3}{2p^{s}} + ⋯) ,

we find that the abscissa of convergence of f_{+}(s) is ≤ β_{0}. Hence, f_{+}(s) converges absolutely in Re s ≥ β_{0} and in particular, for Re s ≥ \frac{1}{2}. Therefore, the singularities of log ϕ(s) coincide with the singularities of f_{−}(s) in Re s ≥ \frac{1}{2}. By a standard method, it is possible to show that if N(T) denotes the number of zeroes of ψ(s) in the critical strip, up to height T, we find

N(T) = \frac{2}{π} T log T + O(T).

As ζ(s) has only \frac{T}{2π} log T + O(T) zeroes in this region, we find that log ϕ(s) has singularities in Re s ≥ \frac{1}{2}. But f_{−}(s) has non-negative Dirichlet coefficients and so, by a famous theorem of Landau, one of these singularities is real. This contradicts the fact that ϕ(s) has no real zeroes. This proves the lemma.

Now we know

∑_{τ_{p}^{2}>1} \frac{τ_{p}^{2} − 1}{p^{β}} = +∞ for β < \frac{1}{2}.

In a standard way, it is now possible to deduce that for some 0 ≤ γ ≤ β,

card (e^{m} < p < e^{m+1} : p^{−γ−ε} < τ_{p}^{2} − 1 < p^{−γ}) ≥ \frac{(e^{m})^{β+γ}}{m^{2}} .

Setting

N = ∏_{p} p

where the primes range over the primes in the above set, we find

τ_{N} ≥ ∏_{p} (1 + \frac{1}{p^{γ+ε}} )

≫ exp (∑_{p} \frac{1}{p^{γ+ε}} )
≫ exp ((log N)^{\frac{β−ε}{β+γ}} ) .

As γ ≤ β, we get τ_{N} = Ω(exp(log N)^{\frac{1}{2} −ε}). By utilizing prime powers, it is possible to improve the exponent \frac{1}{2} to \frac{2}{3}. The key idea is supplied by the lemma:

Lemma 2. There is a constant c > 1 such that for every prime p, there is an m(p) such that

m(p) ≪ \frac{1}{|τ_{p}^{2} − 1|}

and τ_{p^{m(p)}} ≥ c > 1.

We leave the proof of this lemma as an exercise to the reader.

5. Concluding remarks

There is nothing special about Ramanujan’s cusp form, with respect to Ωtheorems. We have:

Theorem 3. If f(z) = ∑_{1}^{∞} a(n)e^{2πinz} is a normalized Hecke eigenform of weight k, then

a(n) = Ω (n^{\frac{k−1}{2}} exp (\frac{c(log n)^{1/k}}{log log n} )) .

If similar results concerning real zeroes of the Rankin convolution were known, this theorem could be improved. We remark that the Sato-Tate conjecture is equivalent to showing that each of the Dirichlet series

∑_{1}^{∞} \frac{τ_{n}^{k}}{n^{s}}

has an analytic continuation for Re s ≥ 1 and no zeroes on σ = 1 for k ≥ 3. At present, this is only known for k = 3.

We have shown that τ(n) becomes very large infinitely often. But there is a conjecture of Elliott [3] that this should not happen too often. More precisely, Elliott showed either

(i) ∑_{n≤x} |τ_{n}| = o(x) as x → ∞

or

(ii) ∑_{p} \frac{(τ_{p}−1)^{2}}{p} < +∞, but not both.

He could not decide which was true. We shall show

∑_{p} \frac{(τ_{p} − 1)^{2}}{p} = +∞.

Consider

g(s) = ζ(s) (∑_{1}^{∞} \frac{τ_{n}^{2}}{n^{s}} ) (∑_{1}^{∞} \frac{τ_{n}}{n^{s}} )^{−2} .

Then g(s) has an analytic continuation to Re s ≥ \frac{1}{2} except for a pole of order 2 at s = 1.

Moreover,

g(s) = ∏_{p} (1 + \frac{(τ_{p} − 1)^{2}}{p^{s}} + ⋯)

and it is easily seen that

log g(s) = ∑_{p} \frac{(τ_{p} − 1)^{2}}{p^{s}} + g_{1}(s)

where g_{1}(s) is analytic for Re s > \frac{1}{2}. Therefore, we find

∑_{p} \frac{(τ_{p} − 1)^{2}}{p^{s}} = 2 log (\frac{1}{s − 1}) + g_{2}(s),

where g_{2}(s) is analytic as s → 1^{+}. Choosing s = 1 + \frac{1}{log x} reveals, in a straightforward manner,

∑_{p≤x} \frac{(τ_{p} − 1)^{2}}{p} ≫ log log x.

In fact, one can show

∑_{p≤x} \frac{(τ_{p} − 1)^{2}}{p} = (2 + \underline{o}(1)) log log x,

by standard methods.

Shortly after this conference, Professor K. Ramachandra suggested that it may be possible to show effectively,

τ_{n} = Ω (exp (\frac{c(log n)^{2/3}}{(log log n)^{A}} ))

for some A > 0, by utilizing the theorem in [11].

By the theorem of Landau, it follows that

∑_{p≤x} (τ_{p}^{2} − 1) log p ≥ cx^{\frac{1}{2}}

for some c > 0. This, however, is ineffective, but can be made effective by an elaborate averaging argument carried out in [11], which we shall not discuss here.

By decomposing the interval [1, x] into O(log x) intervals, it follows that for some u,

∑_{\begin{matrix}u≤p≤2u \\ τ_{p}^{2}>1 \end{matrix}} (τ_{p}^{2} − 1) > \frac{cu^{\frac{1}{2}}}{(log u)^{2}} .

As before, it is easily deduced that for some m ≤ \frac{1}{2} log u,

card (u ≤ p ≤ 2u : e^{−m−1} < τ_{p}^{2} − 1 ≤ e^{−m}) ≥ \frac{cu^{\frac{1}{2}} e^{m}}{(log u)^{3}} .

Proceeding as before, we find finally,

τ_{n} = Ω (exp (\frac{c(log n)^{2/3}}{(log log n)^{5/3}} )) .

Acknowledgments . I would like to thank Professor K. Ramachandra for bringing Elliott’s conjecture to my attention and Matscience (Madras) for their financial support and hospitality during the Conference. R. Balasubramanian and I thank Prof. A. Selberg for suggesting the main ideas which led to our result.

References

[1] R. Balasubramanian, and M. Ram Murty, An Ω-theorem for Ramanujan’s τ-function, to appear.

[2] P. Deligne, La conjecture de Weil, I, Publ. Math. I.H.E.S. 43 (1974), 273–307. (See also, Formes modulaires et representations l-adiques, Springer lecture notes 179 (1971), 139–172.)

[3] P.D.T.A. Elliott, Mean value theorems for multiplicative functions bounded in mean αpower, α ≥ 1, J. Australian Math. Soc. (Series A) 29 (1980), 177–205.

[4] G.H. Hardy, Ramanujan, Cambridge University Press (1940), pp. 178–180.

[5] E. Hecke, Mathematische Werke, Vandenhoeck und Ruprecht, Gottingen (1959).

[6] H. Joris, An Ω-result for coefficients of cusp forms, Mathematika 22 (1975), 12–19.

[7] L.J. Mordell, On Ramanujan’s empirical expansions of modular functions, Proc. Camb. Phil. Soc. 19 (1917), 117–124.

[8] H. Petersson, Konstruktion der Samtlichen losungen einer Riemannachen Funktional gleicheng durch Dirichlet-Reihen mit Eulerschen Produktenwicklung, I, Math. Ann. 116 (1939), 401–412.

[9] S. Ramanujan, On certain arithmetical functions, Trans. Cambr. Phil. Soc. 22 (1916), 159–184.

[10] R.A. Rankin, An Ω-result for coefficients of cusp forms, Math. Ann. 203 (1973), 239–250.

[11] K. Ramachandra, and R. Balasubramanian, Some problems in analytic number theory II, to appear.

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, INDIA