FUNDAMENTAL PROPERTIES OF SYMMETRIC SQUARE L-FUNCTIONS. I

Transcription

1 Illinois Journal of Mathematics Volume 46, Number, Spring 00, Pages 3 43 S FUNDAMENAL PROPERIES OF SYMMERIC SQUARE L-FUNCIONS. I A. SANKARANARAYANAN Dedicated with deep regards to Professor K. Ramachandra on his seventieth birthday Abstract. We improve the existing upper bound for the mean-square of the absolute value of the Rankin-Selberg zeta-function attached to a holomorphic cusp form) defined for the full modular group in the critical strip.. Introduction A remarkable result of Selberg see [57]) says that a positive proportion of zeros of the Riemann zeta-function are on the critical line. Similar results were obtained by Hafner for L-functions attached to cusp forms which are Hecke eigenforms see []). Another important problem is studying the growth of the L-functions under consideration. In this connection, in a celebrated paper [8], Iwaniec and Sarnak proved growth estimates for eigenfunctions of certain arithmetic surfaces which break the bound that can be obtained by convexity arguments. his raises the question of proving non-trivial in the sense of breaking the usual convexity bounds) growth estimates for general L-functions. Of course, this is closely related to the Lindelöf hypothesis. We should also point out here the important work by Iwaniec, and by Duke, Friedlander and Iwaniec see [7], [9], and [0]), who show how one can break the convexity bounds in different aspects, namely Q and r, for certain automorphic L-functions. For an excellent exposition of these results and for further comments we refer to [56]. he problem of studying the difference between consecutive zeros on the critical line was considered by various authors; see, for example, [], [5], [30], [3], and [54]. In [60], Shimura proved that the completed symmetric square L-functions can be continued analytically to entire functions on the whole complex plane by establishing a functional equation. Received September 9, 000; received in final form April 5, Mathematics Subject Classification. Primary F66. Secondary M06, M4. 3 c 00 University of Illinois

2 4 A. SANKARANARAYANAN In recent times, there has been much interest in establishing the analytic continuation and a functional equation for various symmetric power L-functions see [59]). We always write s = σ + it, z = x + iy. Let fz) = a ne πinz be a holomorphic cusp form of even integral weight k defined over the full modular group SL, Z). We assume that a n are eigenvalues of all Hecke operators and a =. Let α p and β p be the complex numbers defined by the equation.) a p p s + p k s = α p p s) β p p s). he Hecke L-function attached to f is defined as.) Ls, f) = a n n s. It is absolutely convergent in a certain half-plane and is continuable analytically to an entire function on the whole plane. For an arbitrary primitive Dirichlet character ψ, the symmetric square L-function attached to f is defined as.3) Ds) := Ds, f, ψ) := ψp)α p p s) ψp)βpp s) ψp)p k s)) p := a n n s. Here a n is just a notation and does not mean the n -th Fourier coefficient of f.) Following the notation in [60], throughout this paper we assume that χ is a Dirichlet character modulo M and the trivial character when M =, and that ψ is an arbitrary primitive Dirichlet character with conductor r, and the trivial character when r =. Now, Ds) converges absolutely in Rs > k. he critical strip for Ds) is k σ k, and the critical line is σ = k /. We also note that from Deligne s work see [7] and [8]) it follows that see the Note added in proof at the end of the paper).4) a n dn)) n k. he following are some fundamental questions about these L-functions:. Is Dk + it) different from zero for all t R?. If the answer to Question is yes, can we establish a reasonable zerofree region for Ds)? 3. Is it possible to establish mean-value theorems on certain lines? 4. If the answers to Questions and are yes, can we prove certain density theorems for the zeros of Ds)?

3 FUNDAMENAL PROPERIES OF SYMMERIC SQUARE L-FUNCIONS. I 5 he answers to Questions and above are known see Lemma 3. in [3] and, for example, Exercise 3.. of [49]). In fact, using the Fundamental Identity Lemma 3.) of this paper combined with Lemma 3. of [3], it is not difficult to establish a reasonable zero-free region. After Shimura s work see [60]), the answers to the above fundamental questions form the basis for any further progress. It should be mentioned here that mean values of derivatives of modular L-series had been studied earlier by Ram Murty and Kumar Murty in [50]. In this paper, we concentrate only on Question 3 above. Results related to Question 4 above will form part II of this paper, which will appear else. he properties of Rankin-Selberg zeta-functions have been studied extensively by many authors; see, for example, [4], [5] and [34]. After normalizing the coefficients, the Rankin-Selberg zeta-function is defined by.5) Zs) = ζs) a nn k s = c nn s say). his Dirichlet series is absolutely convergent in the half plane σ > and can be continued as a meromorphic function to the whole complex plane with a simple pole at s =. It satisfies a nice functional equation see [4]). For example, in [4], Ivic studied mean-value theorems for Zs) for a certain range of σ; from his work it follows that ).6) Z + it dt +ɛ 0 for every ɛ > 0. In [34], Matsumoto proved the following result see heorem of [34])..7).8) heorem A. i) For / σ 3/4 we have for any ɛ > 0. ii) For 3/4 < σ we have 0 Zσ + it) dt 4 4σ log ) +ɛ Zσ + it) dt = c n n σ + O θσ)+ɛ ), 5 σ if 3 4 σ + 9, 0 θσ) = 60 σ) if + 9 σ. 9 0σ 0

4 6 A. SANKARANARAYANAN In particular, heorem A gives ).9) Z + it dt log ) +ɛ, 0 which is a slight improvement of.6). he aim of this paper is twofold. Firstly, we study the analytic properties of the symmetric square L-functions. Secondly, exploring these ideas, we obtain a nice improvement of heorem A. For example, from heorem 4. below it is immediate that unconditionally) ).0) Z + it dt /6+ɛ 0 for every ɛ > 0. We also show see heorem 4.3) that, under the Lindelöf hypothesis for the Riemann zeta-function,.) for every ɛ > 0. 0 Z / + it) dt 3/+ɛ Acknowledgement. he author has great pleasure in thanking Professor Ram Murty for fruitful and stimulating discussions. he author is grateful to the Department of Mathematics and Statistics at Queens University for its kind invitation and warm hospitality which enabled him to complete this project. he author is highly indebted to the anonymous referee for his valuable comments.. Notation and preliminaries he letters C and A with or without suffixes) denote effective positive constants unless otherwise specified. he constants need not be the same at every occurrence. hroughout the paper we assume 0, 0 is a large positive constant. We write fx) gx), or fx) = Ogx)), to mean that fx) < C gx). All implied constants are effective. We set s = σ + it and w = u + iv. In any fixed strip a σ b we have, as t,.) Γσ + it) = t σ+it / e π/ it+iπ/)σ /) π Let.) Rs) = π 3s/ Γ s ) Γ s + ) Γ s k + hen Ds) satisfies the functional equation see [60]).3) Rs) = Rk s). + O ) Ds). )). t

5 FUNDAMENAL PROPERIES OF SYMMERIC SQUARE L-FUNCIONS. I 7 Also we note that if ) s k +.4) R s) = π s k+)/ Γ ζs k + ), then ζs k + ) satisfies the functional equation.5) R s) = R k s). herefore if D s) = ζs k + )Ds), then D s) satisfies the functional equation.6) Rs)R s) = Rk s)r k s), and we see that Rs)R s) extends D s) to an analytic function in the whole plane except for a simple pole at s = k. We define.7) ξs) = s k)k s k)rs)r s). Note that.8) ξs) = ξk s). We write.9) Ds) = χs)dk s), ) ) ) k s k s k s + Γ Γ Γ.0) χs) = π 3k ) +3s ) ). s s + s k + Γ Γ Γ ) From.) and.0) it follows that, for a σ b, we have as t, ) 3it )).) χs) = C k, σ)t t 6k 6σ 3) + O, πe t C is a certain constant depending only on k and σ. From the maximummodulus principle and the functional equation, we obtain.) Dσ + it) t 3 k σ) log t uniformly for k / σ k, t Some lemmas Lemma 3. Fundamental Identity). We have fs) := a nn s = ζ s k + ) ζs k + ) Ψs), Ψs) = p + p k s a pp s + p k s).

6 8 A. SANKARANARAYANAN Proof. We note that a n is multiplicative and satisfy the equations 3.) a p λ = a p a p λ p k a p λ and 3.) p k a p λ 3 = a p λ + a p a p λ. From this we deduce first 3.3) fs) = p + a pp s + a p p s + a p j p js j=3 Next, computing the expression 3.) p k 3.) gives the relation 3.4) a p a λ p p k ) a p + λ pk a p p k ) a p λ p3k ) a pλ 3 = 0. Here 3.5) a p j p js = a p p k )a p p k a j p p k )a p + p 3k ) a j p j 3 p js j=3 j=3 a = a p p k )p s p a j p k a p p k )p s p j p js j=. p js j= + p 3k ) 3s = a p p k )p s p k s a p p k ) + p 3k ) 3s) a p j p js j=0 a p j p js j=0 a p p k )p s a pa p p k )p s + a p p k )p k s. We also find that + a pp s + a p p s a p p k )p s a pa p p k )p s + a p p k )p k s = + p k s. Let X = + a pp s + a p p s + a p j p js j=3

7 FUNDAMENAL PROPERIES OF SYMMERIC SQUARE L-FUNCIONS. I 9 and Y = a p p k )p s p k s a p p k ) + p 3k ) 3s. From 3.4), 3.5), and the above arguments, we observe that 3.6) X = XY + + p k s. herefore we obtain fs) = ) + p k s 3.7) a p p p k )p s + p k s a p p k ) p 3k ) 3s = ) p k s + p k s + pk s a pp s + p k ) s ) p = ζ s k + ) ζs k + ) Ψs), 3.8) Ψs) = p his proves the lemma. + p k s a pp s + p k ) s). Remark. his lemma is essentially due to Ramanujan. For a proof of the above lemma in the case of Dirichlet series attached to Ramanujan s τ function see, for example, [5] and [53]. Lemma 3. Montgomery-Vaughan). If h n is an infinite sequence of complex numbers such that n h n is convergent, then +H h n n it dt = h n H + On)). Proof. See, for example, Lemma 3.3 of [39], or [46]. 4. Mean-square upper bounds on certain lines K. Chandrasekharan and R. Narasimhan [4] showed that, whenever a Dirichlet series has a functional equation, an approximate functional equation holds, which has a nice form provided the coefficients of the Dirichlet series are positive. his result can be used to study mean value theorems. Even if the coefficients are not positive, such an application is still possible in some special cases. K. Ramachandra observed that using only the functional equation one can prove reasonable upper bounds for mean-values on certain lines, and he has used this idea in many of his papers see, for example, [43] and [44]). In this section we use the same idea to prove the following result.

8 30 A. SANKARANARAYANAN i) ii) iii) heorem 4.. For t, 0, we have Ds ) dt 3/ log ) 7, Ds ) dt = Ds 3 ) dt = a n n + O 3/4 log ) 7), k / a n n k+ log + O log ) 5), s = k / + it, s = k /4 + it, s 3 = k + / log ) + it, and the implied constants depend on the weight k. Proof of i). Let Y and Y be two parameters satisfying 0 Y, Y A, to be chosen appropriately later. Let ɛ = log ). By Mellin s transformation, we have 4.) S := = πi a n e n/y s n Rw=/+ɛ v log ) = Ds ) + πi = Ds ) + I + O Ds + w)y w Γw)dw + O Y +ɛ Clog e )) Rw= +ɛ v log ) Ds + w)y w Γw)dw + O Y +ɛ e Clog )) + O Y +ɛ e Clog )) + O Y +ɛ C Clog e )) Y +ɛ C e Clog )), say, upon moving the line of integration to Rw = + ɛ. Now, I = 4.) Ds + w)y w Γw)dw πi = πi = I + I, Q = Rw= +ɛ v log ) Rw= +ɛ v log ) n Y a n n s+w k+, Q = χs + w) Q + Q ) Y w Γw)dw n>y a n n s+w k+.

9 FUNDAMENAL PROPERIES OF SYMMERIC SQUARE L-FUNCIONS. I 3 We note that in I, Rs + w) = k 3/) + ɛ. We have 4.3) a n n s+w k+ = a n n s+w k+ n>y Y <n Y 0 + O dn)) n k +k 3 +ɛ k+ = Y <n Y 0 n>y 0 a n n s+w k+ + O Y 5+0ɛ we have used inequality.4). Using Hölder s inequality and a theorem of Montgomery and Vaughan see [39]), we get 4.4) I dt log ) 6+0ɛ Y Rw= +ɛ v log ) log ) 6+0ɛ Y χs + w) Y 0 n>y a nn s+w k+ +log ) log ) U log ) 6+0ɛ Y log ) 6+0ɛ Y U C log Y j=0 U n U U U n U ) Y w Γw)dw dt a n n k k 3 +ɛ)) n dn)) 4 n k n n k k 3 +ɛ)) U n U log Y ) 5 j Y ) +4ɛ ), + 7+0ɛ Y Y 0+40ɛ a n n s+rw k+ dt + 7+0ɛ Y Y 0+40ɛ + 7+0ɛ Y Y 0+40ɛ + 7+0ɛ Y Y 0+40ɛ + 7+0ɛ Y Y 0+40ɛ log ) 6+0ɛ Y Y +4ɛ log Y ) ɛ Y Y 0+40ɛ.

10 3 A. SANKARANARAYANAN Also we have 4.5) I = πi = πi Rw= +ɛ v log ) Rw= v log ) χs + w) a n n s+w k+ Y w Γw)dw n Y χs + w) a n n s+w k+ Y w Γw)dw n Y + O C e C log ) ), upon moving the line of integration to Rw =. Using again the Montgomery-Vaughan theorem and inequality.4), we obtain 4.6) Now, and so 4.7) I dt log ) 3 Y log ) 3 Y log ) 3 Y log ) 3 S = +log ) log ) n Y n Y Y log Y ) 5. a n n s+rw k+ dt n Y a n n n k k )) dn))4nk n k k )) a n n s e n/y, S a dt = n n k /) e n/y + On)) a = n n )) n k + O Y n Y/ + O a n n k n Y/ Y n ) α+

11 FUNDAMENAL PROPERIES OF SYMMERIC SQUARE L-FUNCIONS. I 33 + O n Y/ a n n k n + n Y/ a n ) α+ Y n k n n log Y ) 5 + Y log Y ) 5 with α = k. Using inequality.4), we find that 4.8) 4.9) Y n Y/ n Y/ dn)) 4 n k nk log Y ) 5, dn)) 4 n k nk n log Y ) 5. Choosing α = k, we have 4.0) Y k n Y/ dn)) 4 n k nk n k Y k Y k log Y ) 5, j=0 U n<u, U= j Y/ j=0 U= j Y/ dn)) 4 n k Ulog U)5 U k 4.) n Y/ dn)) 4 n k nk n Y log Y ) 5. Similarly to 4.0), we obtain, by choosing a suitable α say, α = k) 4.) n Y/ dn)) 4 ) α+ Y nk Y log Y ) 5. nk n Part i) of the theorem now follows if we choose Y = Y = 3/. Proof of ii). Let Y and Y be two parameters satisfying 0 Y, Y A, which will be chosen appropriately later. Let ɛ = log ). Since the proof of ii) is similar to that of i), we only give a sketch. By Mellin s

12 34 A. SANKARANARAYANAN transformation, we have 4.3) S := = πi a n e n/y s n Ds + w)y w Γw)dw + O Y +ɛ Clog e )) Rw=/+ɛ v log ) = Ds ) + πi Rw= +ɛ v log ) Ds + w)y w Γw)dw + O Y +ɛ Clog e )) + O Y +ɛ C Clog e )) = Ds ) + I + O Y +ɛ e Clog )) + O Y +ɛ C e Clog )), say, upon moving the line of integration to Rw = + ɛ. Now, 4.4) I = πi = I + I, Rw= +ɛ v log ) χs + w) Q 3 + Q 4 ) Y w Γw)dw Q 3 = a n n s+w k+, Q 4 = a n n s+w k+. n Y n>y We note that in I, Rs + w) = k 5/4) + ɛ. We have 4.5) n>y a n n s+w k+ = Y <n Y 0 a n n s+w k+ + O ) Y.5+0ɛ, we have used inequality.4). Using Hölder s inequality and a theorem of Montgomery and Vaughan see [39]), we get as in the proof of i) 4.6) I dt log ) 4.5+0ɛ Y Y +4ɛ log Y ) ɛ Y Y 5+40ɛ.

13 FUNDAMENAL PROPERIES OF SYMMERIC SQUARE L-FUNCIONS. I 35 Also, we have 4.7) I = πi Rw= 3 4 v log ) χs + w) a n n s+w k+ Y w Γw)dw n Y + O C e C log )), upon moving the line of integration to Rw = 3/4. Using again the Montgomery-Vaughan theorem and inequality.4), we obtain I dt log ) 3 +log ) 4.8) Y 3/ a n n s+rw k+ dt log ) n Y Now, log ) 3 Y 3/ log Y ) 5. S = and, as in the proof of i), we obtain 4.9) S dt = a n n s e n/y, a n n k /4) + O Y / log Y ) 5 + Y / log Y ) 5). Part ii) of the theorem now follows by choosing Y = Y = 3/. Proof of iii). Part iii) of the theorem follows from Lemma 3.. We are now in a position to prove the following result. heorem 4.. Suppose that the inequality ) ζ + it t κ log t) holds for some κ > 0 and all t 0. hen: i) For / σ 3/4 we have Zσ + it) dt +4κ+) σ) log ) 73 4σ)+.

14 36 A. SANKARANARAYANAN ii) Let and ν = 5 6 4κ) ϑ = 9 0ν + 00ν + 40ν For 3/4 < σ we have Zσ + it) dt = c n n σ + O θσ)+ɛ ), 5 σ if 3 4 < σ ϑ, θσ) = 30 σ) if ϑ σ. 7 5ν 0σ Proof of i). From Lemma 3., we have the relation Zs k + ) = ζs k + )Ds). After normalizing the coefficients we find that 4.0) Zs) = ζs)ds + k ). Let ) λ 4.) Jσ, λ) = fσ + it) /λ, λ > 0. hen Gabriel s convexity theorem see p. 03 of [6]) asserts that for α σ β 4.) Jσ, pλ + qµ) J p α, λ)j q β, µ), 4.3) p = β σ β α, We choose the parameters as follows: q = σ α β α. fs) = Ds + k ), α =, β = 3 4, λ =, µ =. hen p + q = and pλ + qµ = /. From i) and ii) of heorem 4., using 4.), we obtain, for / σ 3/4, / D σ + k + it) dt) Q 5,

15 FUNDAMENAL PROPERIES OF SYMMERIC SQUARE L-FUNCIONS. I 37 ) Q 5 = D k ) 3 4σ) + it dt D k ) 4 + it dt his implies that 4.4) Dσ + k + it) dt 3/ log ) 7) 3 4σ 4σ 5 4σ)/ log ) 73 4σ). From the assumption of the theorem it follows that for / σ 4.5) ζσ + it) t κ σ) log t. Also, notice that 4.6) ) 4σ ). Zσ + it) dt ) ) max ζσ + it) t Dσ + k + it) dt. From 4.4), 4.5), and 4.6), part i) of the theorem follows. Proof of ii). We only sketch the proof, since the details can be found in [34]. he function Zs) satisfies the functional equation 4.7) Zs) = s)z s), 4.8) s) = Γ s)γk s) π)4s t Γs)Γs + k ) π ) 4σ + O )) t for t t 0. From the definition of ν, we have 5/ < ν < /. herefore, from 4.8) and part i) of heorem 4., we find that 4.9) If we write Zν + it) dt t 4 8ν Z ν + it) dt 4 8ν+ +4κ+)ν)+ɛ +ɛ. c n = Cx + x) n x then, combining 4.9) with the method of Lemma 3. of [3], we obtain 4.30) X y) dy X +ν+ɛ.

16 38 A. SANKARANARAYANAN Following the arguments of [34] and using 4.30), ii) follows. Remark. Using the classical value κ = /6, heorem 4. gives ) Z + it dt /6 log ) 7, which is far better than.9). he best known value for κ is κ = ɛ see []). We also mention that part i) of heorem 4. improves upon part i) of heorem A only in the range / σ 3 8κ)/4 8κ), since + 4κ + ) σ) < 4 4σ holds only when σ 3 8κ)/4 8κ) < 3/4. heorem 4.3. constant ɛ, If the Lindelöf hypothesis holds, i.e., if for every positive ) ζ + it t ɛ, then the inequality Zσ + it) dt + σ)+ɛ holds for every positive constant ɛ in the range / σ 3/4 ɛ. Proof. his follows from part i) of heorem 4. upon taking κ = ɛ. heorem 4.4. Assume that ) j ζ + it dt +ɛ holds for some fixed integer j and every positive constant ɛ. hen we have ) Z + it dt 3/)+/j)+ɛ. for every ɛ > 0.

17 FUNDAMENAL PROPERIES OF SYMMERIC SQUARE L-FUNCIONS. I 39 Proof. From 4.0), on using Hölder s inequality, we obtain 4.3) ) Z + it dt ) = ζ + it D k ) + it dt ) ) j /j ζ + it dt D k ) j/j ) + it dt max t D k ) ) /j ) + it j ζ + it dt D k ) j )/j dt) + it. ) /j ) j )/j By.), we have 4.3) Dk + it) t3/4 log t). he asserted estimate now follows from the assumption of the theorem, part i) of heorem 4., 4.3), and 4.3). Remark. From heorems 4., 4.3 and 4.4, we see that the mean-square upper bound for Z/ + it) depends on ) the growth estimate of the Riemann zeta-function on the line σ =, ) the higher moments of the Riemann zeta-function on the critical line σ =, 3) the growth estimate of the symmetric square L-function Ds) on the line σ = k, and 4) the higher moments of the symmetric square L-function Ds) on the line σ = k. In this connection we would like to point out the important work of Heath- Brown see [5]) and of Ivic and Motohashi see [6]). On the one hand, a general theorem of Ramachandra see [47]) implies the mean-square lower bound 4.33) ) Z + it dt log log log.

18 40 A. SANKARANARAYANAN On the other hand, it is not hard to see that, under the assumption of the Lindelöf hypothesis for Ds) on the line σ = k, i.e., the estimate 4.34) D k ) + it t ɛ for any small positive constant ɛ, we have ) 4.35) Z + it dt +0ɛ. his suggests the following conjecture. Conjecture. 4.36) with some positive constant A. We have ) Z + it dt log ) A Note added in proof. In fact, we have ) Ds) := ζs k + ) a n n s, a n dn )n k. From our definition.3), it follows that b n := a n = l m=n l k a m and that b n is a multiplicative function. It is not difficult to see that b p j p jk ) j a + ) a=0 b+a=j p jk ) j + ) = p jk ) d p j)) since for every fixed j there is at most one solution to the equation b + a = j for every fixed a. For j 4 the inequality follows by an easy computation, and for j 5 it can be proved by considering separately the cases when j is even and odd.) his implies in our notation) the final inequalty of.4), namely a n dn)) n k.

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21 FUNDAMENAL PROPERIES OF SYMMERIC SQUARE L-FUNCIONS. I 43 [53], Contributions to the theory of Ramanujan s function τn) and similar arithmetical functions. II, Proc. Cambridge Phil. Soc ), [54] A. Sankaranarayanan, Zeros of quadratic zeta-functions on the critical line, Acta. Arith ), 38. [55] P. Sarnak, Some applications of modular forms, Cambridge racts in Math., vol. 99, Cambridge Univ. Press, Cambridge, 990. [56], Arithmetic quantum chaos, he Schur lectures el Aviv, 99), Bar-Ilan Univ., Ramat Gan, 995, pp [57] A. Selberg, On the zeros of the Riemann zeta-function, in: Collected papers, vol. I, Springer-Verlag, 989, pp [58], Contributions to the theory of the Riemann zeta-function, in: Collected papers, vol. I, Springer-Verlag, 989, pp [59] F. Shahidi, On certain L-functions, Amer. J. Math ), [60] G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc ), [6] E. C. itchmarsh, he theory of the Riemann zeta-function. nd edition, Clarendon Press, Oxford, 986. Department of Mathematics and Statistics, Queen s university, Kingston, Ontario, Canada K7L 3N6 address: sank@mast.queensu.ca Current address: School of Mathematics, ata Institute of Fundamental Research, Homi Bhabha Road, Mumbai , India address: sank@math.tifr.res.in