The Fourth Moment Of Ramanujan T-function

IMlemt lle Am

Math. Ann. 266, 233-239 (1983)

O Springer-Verlag1983

The Fourth Moment of Ramanujan z-Function Carlos J. Moreno 1 and Freydoon Shahidi 2'* 1 Department of Mathematics, University of Illinois, Urbana, [L 61801, USA 2 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

The purpose of this note is to give a proof of the following result for the Ramanujan arithmetical function z(n) = %(n)n 11/2 Main Thenrem. The Dirichlet series F(s)= ~ Zo(n)4n -s n=l

has a meromorphic continuation to the half plane Re(s)_->~ and in the region Re(s)> 1 it is holomorphic except for a double pole at s = l . In particular .for a positive real number x we have to(n)4 ~ cx logx, n~x

where c is a positive constant. The interest in establishing a result of this nature comes from some recent work of Rankin concerning non-trivial estimates for the higher moments of Zo(n) and other arithmetical functions [-4], and specially from his applications to a result of Elliott about mean values of multiplicative arithmetic functions I-1]. The proof we give is based on the work of the second author on the L-function L(s, re, Sym4Q) applied to the automorphic representation n=rc d corresponding to the Rarnanujan modular form

FI (l-q")24= 2

n=l

n=l

and especially the non-vanishing of these L-functions on the line Re(s)= 1. These ideas are in turn an outgrowth of the important work of Jacquet, Piatetski-Shapiro and Shalika on GL,. In Sect. 1 we relate the study of the Dirichlet series F(s) to the functions L(s, red, SymeQ) and L(s, ~za,Sym4Q). In Sect. 2 we establish the analytic properties of F(s) and show how the non-vanishing of L ( l + i t , nA, Sym4o) implies the presence of a double pole for F(s) at s = 1. *

Partially supported by NSF grant MCS81-01600

234

C.J. Moreno and F. Shahidi

1. The Euler Product of

F(s)

In accordance with Deligne's theorem, [z(p)l<2p 1~/2, the roots of the equation 1 - z(p)p- ~X/ZT+ T 2 = 1 - Zo(p)T+ T 2 are complex conjugate and hence we can put for each prime p

Zo(p) = ~p + ~;1 where ~v is a complex number of absolute value 2. We let 0 :GL2(C)~GL2(~) denote the standard two dimensional representation. In the notation of [7], with

n , = n ~ | (@nv)being the automorphic representation of GL2(~k_Q)which corresponds to the Ramanujan modular form, we put

L(s, n~, Sym 1~) = F(s, n~, Sym 1O)~l(s), L(s, rG, Sym2 O) = F(s, n~, Sym2 Q)ff2(s), L(s, rG, Symr

= F(s, n~, Sym4 0)~,~(s),

where F(s, n~, Sym ~0) is a product of F-functions which we need not specify and where ~ ( s ) = 1-I {(1 - ~,p-~)(1 - ~ ; ~ p - ~ ) } - ' , P

P

and ~4(s) = I ] / ( 2 - ~ e - ' ) ( 2

- ~v-

2)(2 - p-~)(1 - ~ ; 2 p - ' ) 0

- ~ ; 4p-~)} - ~.

P

In particular we have

L ( s - ~ , n~, Sym 1a) = (2n)- SF(s) l-[ (1 - z(p)p-~ + p tl - z,)- 1 P

=(2n)-SF(s) ~ z(n)n -~. n= 1

Since zo(n)=z(n)n -11Iz is bounded in absolute value by d(n), the number of positive divisors of n, we can obtain by elementary means fairly precise information about F(s) within the region of absolute convergence.

Lena

1. For R e ( s ) = o > 1 one has r ~6 If(s)l < ~(T~)~i.

Proof. From the multiplicative property of Zo(n) and the fact that d(p ~)= v + 1 we obtain

Ramanujan r-Function

235

If(s)l =

z0 p~ v=

<

zo

=

V

4-

P

--vo"

i;=

=<

v 4 p - va tJ=

From the identity

I+llT+llT2+T3 (l-T) s

~, (V_.k 1)4 TV

~:o and the inequality 1 + llT+

llTZ+

T3=<(1 + T) ~ -

(1 -

T 2) 11

(1 - T) ll '

with T = p - ~ we get the lemma. In the following (o(S) -- ((s) will denote the Riemann zeta function. Lemma 2. With notations as above and for Re(s)> 1, one has

Zo(n)4n-" = ~(s)Z~2(s)3~4(s)1-I Lp(p-~), n=l

p

where Lp(T) is a polynomial of degree 14 whose coefficients are bounded by an absolute constant independent of p and whose constant and highest term equals 1. Proof. F r o m the multiplicativity property of Zo(n)4 we have the Euler product identity ro(n),~n-S = i'l= i

Vo(pV)4p-W. U=

Therefore, the claim will follow if we prove, with T = p -~, that

v=0

where for g___ ( ~

~l)

z~

Lp(T) = (1 - T)ESE(T)aS4(T) '

w e h a v e put

S2(T) = det(l 3 - Sym 2 Q(g)T) ---(1 - r

T)(1 - r

S4(T) = det (Is - SYm4 Q(g) T) = ( 1 - (~ T)(1 - (2 T)(1 - T)(1 - ~- 2T)(1 - ~ 4 T ) ,

236

C.J. Moreno and F. Shahidi

and Lp(T) is a polynomial satisfying the properties of the lemma. To simplify notation we put ~ = ~p. F r o m the basic recurrence relation for the eigenvalues of the Hecke operators, which can be stated in terms of the formal identity 1

1 - Zo(P)T+ T 2

= )_2 ro(F') r~', ~=o

or equivalently as

(I-r

~o,-(_--~=5-]

r~,

we see that our problem is reduced to showing that [~v+_l v=O \

Lp(r)

~-v-1)4

~ - ~- ~

9 7" - (l -

T)2S2(T)3S4(T)

By a straightforward calculation one shows that

/a~+~ 1\4 v_~o/~l)

tv=

l+a(3+5a+3a2)t+a3(3+5a+3a2)t2+a6t 3 (1--a4t)(1--a3t)(1--a2t)(1--at)(1--t)

T Substituting in this identity a = ~z and t = ~ - , and using the fact that

a(3+5a+3a2)t=( 2 ( 3 + 5 ( 2 + 3 ( 4) ~T= (3Zo(p) 2 - 1)T, we obtain ~ z0(pV)4T~ - N(T) =o S4(T)' where

N(T) = 1 + (3Zo(p) 2 - 1 ) T + (3Zo(p) 2 - 1)T 2 + T 3 9 If we make use of the fact that

Sz(T ) =

1 - (Zo(p) 2 - 1)T + ('Co(p) 2 - 1 ) T 2 -+- T 3 ,

we obtain that

Lp( T) = =

N ( T)S2( T)3(1 -

T)2

1 - c2(P)T 2 +..,-}-

T 14 ,

is a polynomial of degree 14 whose coefficients are b o u n d e d by an absolute constant independent of p, whose constant and highest term equal 1, and whose linear term equals 0. This completes the p r o o f of the lemma.

Ramanujan r-Function

237

2. Analytic Properties of the Function F(s) The meromorphic continuation of the Euler product L(s, ~z~,Sym 2Q) = F(s, 7zoo,Sym 2 O)(2(s), where F(s,~zo~,Sym2Q)=Fm(s+ ll)Fm(s+ 12)Fro(s+ 1), and Fm(s)=Tz-S/2F(2),

was

established by Rankin [5] who also proved that (2(1 +it)@O for all real t. That L(s,~zd, Sym2~o) is actually an entire function was first proved by Shimura ([-8], Theorem 1). This was generalized by Jacquet and Gelbart for all automorphic representations ~ of GL 2 over any global field [9]. From the convergence of the series ~ p - 2 S for all s with Re(s)>89 and the p

information we have about the polynomial that the Euler product

Lp(T) given

by Lemma 2, we obtain

~I Lp(p -~) p

converges uniformly for any e > 0 in the region Re(s) >__~+ e ; furthermore in this region it represents a holomorphic function which is free of zeros. To complete the proof of the main theorem we now prove the following result. Theorem 1. The function L(s, ~zn,Sym4Q) has a meromorphic continuation to the whole s-plane, which is holomorphic in the region Re(s)>__ 1 and satisfies (i) L(s, z~A, Sym40) = L(1 - s, 7z~,Sym40) ; (ii) L(1 + it, ~za,Sym4Q) :#0 for all real t.

2. The function L(s, ~a, Sym~ Q)L(s, 7~A,Sym2 Q)L(s, ~zn,Sym4 O) is holomorphic in the whole s-plane except for simple poles at s = 1 and 0, and is free of zeros outside the strip 0 < Re (s)< i. Proof. In [7], Theorem 4.1.1 and Theorem 5.3, it has been shown that for any automorphic representation ~ of PGL(2) which is not monomial, the L-function L(s, ~, Sym40) has a meromorphic continuation to the whole s-plane, satisfies a functional equation of the type (i) and does not vanish on the line Re(s)= 1, except possibly for at most a simple zero at s = 1. Hence it remains to show that it actually has no zero at s = 1 and that in the half plane Re(s)> 1 it is free of poles. This will follow from the following auxiliary lemma, where we use the notation of [2] and [7]. Let S be the finite set of ramified primes including the infinite ones and put

L~ts, ~) = [I L~(s, ~). yes

Lemma 3. Suppose ~ is an automorphic representation of GL2(~k) of a 91obal field

K which is not monomial (i.e. ~| for no Hecke character of K • Then Ls(s, ~, Sym40) is holomorphic in the region Re(s)~ 1 and does not vanish on the line Re(s) = 1.

238

C.J. Moreno and F. Shahidi

Proof of Lemma 3. We shall use the method of Deligne and Gelbart [2]. L e t / / b e the lift of n to PGL3(Zkr) as defined by Gelbart and Jacquet [9]. Then H is cuspidal and Ls(s, H x H) = Ls(s , rr, Sym 2 ~ | Sym 2 ~) = Ls(s, re, Sym 4 Q)Ls(s, re, Sym 2 Q)Ls(s, ~, Sym ~Q), where Ls(s, n, Sym~ is the partial Hecke L-function attached to S. Ls(s, rt, Sym~ has a simple pole at s = l and otherwise is non-zero in the region Re(s)__>l. If L(s, 17 x H) denotes the full L-function of Jacquet, Piatetski-Shapiro, and Shalika on GL(3) x GL(3), then we know it has a simple pole at s = 1 (II~-H). Moreover, the local factors L(s,H~ x 17~), for v~S, are all holomorphic and non-zero in the region Re(s)>1 ([3], Proposition 1.5, p. 507 and Proposition 3.17, p. 542). Consequently, Ls(s, H x 17) has a simple pole at s = 1 and otherwise is non-zero on the line Re(s)= 1 by [7]. This completes the proof of the auxiliary lemma and hence also of the previous theorem. F r o m the basic identity of Lemma 2

Zo(n)'n-S = ~(s)2~2(s)3~4(s) I-[ Le(P-s), n:l

p

and from the previous theorem applied to n = nz, we obtain that the expression on the right hand side has a meromorphic continuation to the half plane Re(s)__>1, at s = 1 it has a double pole, and otherwise it is a holomorphic function free of zeros in the region R e ( s ) > l . By a standard use of the Wiener-Ikehara Theorem we obtain for positive real x sufficiently large Zo(n)4 ~ cx logx, n~x

and s

Z'o(p) 4 logp,-~ 2x.

p<x

This completes the proof of the Main Theorem.

Remarks. The meromorphic continuation of F(s) to the region Re(s)> 88 can be obtained by a more careful analysis of the Euler product l-I Lp(p- s) using the fact P

that

Lp(T) = 1 - (7 - 12Vo(p)2 + 6Zo(p)4)T2 + ... + T 14 . This is of some interest when trying to locate the possible singularities of F(s) to the left of Re(s) = 1. Such information would lead to an asymptotic estimate with an error term O(x4JS(logx)a), as in Rankin's paper [5]. This would then have applications to the problem of the Petersson-Ramanujan conjecture for the Fourier coefficients of the real analytic cusp forms of Maass, namely to

Ramanujan z-Function

239

ra(p)l~p ~ (e>O), w h i c h is the best e s t i m a t e p r e s e n t l y k n o w n for the coefficients o f real a n a l y t i c c u s p f o r m s ; we d o n o t p u r s u e this line h e r e b e c a u s e it would l e n g t h e n the p a p e r d r a m a t i c a l l y . T h e best e s t i m a t e for the P e t e r s s o n R a m a n u j a n c o n j e c t u r e at infinity follows m o r e d i r e c t l y f r o m the w o r k of the second a u t h o r c o n c e r n i n g the L - f u n c t i o n s L(s, ~, S y m 5 Q).

References 1. Elliott, P.D.T.A. : Multiplicative functions and Ramanujan's z-function. J. Austral. Math. Soc. Ser. A 30, 461-468 (1981) 2. Gelbart, S.: Automorphic forms and Artin's conjecture, Lecture Notes in Math., Vol. 627, pp. 241-270. Berlin, Heidelberg, New York: Springer 1977 3. Jacquet, H., Shalika, J.A. : On Euler products and the classification of automorphic representations I. Am. Math. 103, 499-558 (1981) 4. Rankin, R.A. : Sums of powers of cusp form coefficients. Math. Ann. 263, 227-236 (1983) 5. Rankin, R.A.: Contributions to the theory of Ramanujan's function z(n) and similar arithmetic functions, I, II. Proc. Cambridge Phil. Soc. 35, 351-372 (1939); Proc. Cambridge Phil. Soc. 36, 150-151 (1940) 6. Shahidi, F. : On nonvanishing of L-functions. Bull. Am. Soc. Math. 2, 462-464 (1980) 7. Shahidi, F. : On certain L-functions. Am. J. Math. i03, 297-355 (1981) 8. Shimura, G. : On the holomorphy of certain Dirichlet series. Proc. London Math. Soc. 31, 79-98 (1975) 9. Gelbart, S., Jacquet, H. : A relation between automorphic forms on GL(2) and GL(3). Proc. Nat. Acad. Sci. USA 73, 3348-3350 (1976)

Received May 2, 1983; in revised form August 15, 1983