GORAN DJANKOVIĆ AND RIZWANUR KHAN

Transcription

1 A CONJECTURE FOR THE REGULARIZED FOURTH MOMENT OF EISENSTEIN SERIES GORAN DJANKOVIĆ AND RIZWANUR KHAN Abstract. We formulate a version of the Random Wave Conjecture for the fourth moment of Eisenstein series which is based on Zagier s regularized inner product. We prove an asymptotic formula expressing the regularized fourth moment as a mean value of L-functions. This is an advantage over previous work in the literature, which has approached the fourth moment problem through truncated Eisenstein series and not yielded a suitable expression in terms of L-functions.. Introduction One of the main research themes in recent years in the theory of automorphic forms is the problem of mass distribution. Let X =, where H is the upper half complex plane and Γ = SL (Z). In his PhD thesis, Spinu [Sp obtained the following type of weak equidistribution result: (.) E A (z, + it ) 4 dµz T ɛ, X where dµ(z) = dxdy y and E A (z, s) is the truncated Eisenstein series, which on the fundamental domain equals E(z, s) for Im(z) A, and E(z, s) minus its constant term for Im(z) > A. See the next section for a more careful definition. Spinu s result (see also [Lu for a closely related result) is in line with a much more general conjecture, called the Random Wave Conjecture. This conjecture was made for Eisenstein series in [HR, section 7.3. In terms of moments this implies: for any even integer p and any nice compact Ω X, we should have (.) lim T vol(ω) Ω E A(z, + it ) p c p dµz = log T vol(x), p/ where c p is the pth moment of the normal distribution N (, ). The same conjecture is also made for E(z, + it ). As we will see below, log T roughly equals E A (, + it ). One would of course like to go beyond Spinu s upper bound and prove an asymptotic for the fourth moment of Eisenstein series. In [BK, this was achieved, conditional on the Generalized Lindelöf Hypothesis, for Hecke Maass forms of large eigenvalue when Ω = X, and agreement was found with the RWC. Thus in analogy one would expect (.) to also hold for p = 4 and Ω = X, and one may hope that the statement in this case can be proven unconditionally. After all, such problems can be a bit easier for Eisenstein series for example, recall that the case p = of (.) was first proven for Eisenstein series [LS before the analogue was proven for Hecke Maass forms [Li, So. What would the proof of such an asymptotic entail? The starting point in [BK is to relate the fourth moment of an L -normalized Hecke Maass form f to L-functions. One uses the spectral Mathematics Subject Classification. Primary: F, M99; Secondary: 8Q5. Key words and phrases. Automorphic forms, Eisenstein series, equidistribution, L 4 -norm, regularized inner products, quantum chaos, random wave conjecture, L-functions. The first author was partially supported by Ministry of Education, Science and Technological Development of Republic of Serbia, Project no. 748.

2 GORAN DJANKOVIĆ AND RIZWANUR KHAN decomposition and Plancherel s theorem to write (.3) f, f = j f, u j +..., where the inner product is the Petersson inner product, {u j : j } is an orthonormal basis of Hecke Maass forms, and the ellipsis denotes the contribution of the Eisenstein spectrum and constant eigenfunction. Next one can use Watson s triple product formula to relate the squares of the inner products on the right hand side to central values of L-functions. Thus the problem is reduced to one of obtaining a mean value of L-functions. If one tries to mimic this set up for E(z, +it ) in place of f, the first obvious difficulty encountered is that the left hand side of (.3) does not even converge. To circumvent this, Spinu worked with the truncated E A (z, + it ), which decays exponentially at the cusp. However a major drawback is that E A (z, + it ) is not automorphic, so Spinu could not obtain a precise relationship with L-functions. He could only obtain an upper bound [Sp, section 4.. The goal of this paper is to reformulate entirely the fourth moment problem for Eisenstein series. To make sense of E (, + it ), E (, + it ), we contend that it is more natural to use Zagier s regularized inner product [Za, which does converge. The basic idea of Zagier s method is that to kill off the growth of an automorphic form, one should not subtract off the constant term like Spinu does, but rather subtract off another Eisenstein series in such a way that the final object is square integrable and automorphic. This way we will end up with a precise relationship between a regularized fourth moment and L-functions. This is the first goal of our paper, and we will prove Theorem. (Regularized fourth moment in terms of L-functions). Let {u j : j } denote an orthonormal basis of even and odd Hecke Maass cusp forms for Γ, ordered by Laplacian eigenvalue 4 + it j, and let Λ(s, u j) denote the corresponding completed L-functions. Let (s) denote the completed Riemann ζ function. As T, we have for any ɛ >. = 4 π log T + j reg X E(z, / + it ) 4 dµ(z) cosh(πt j ) Λ( + T i, u j) Λ (, u j) L(, sym u j ) ( + T i) 4 + O(log5/3+ɛ T ), This result is potentially very useful. We could try to obtain an asymptotic for the mean value of L-functions on the right hand side (and we will return to this problem in a future paper), thereby obtaining an asymptotic for the regularized fourth moment. This would be nice, but how would we know whether or not our answer is in agreement with the RWC? Thus the purpose of our second result is to translate the RWC to the setting of the regularized fourth moment. As defined in the next section, D A is the part of the fundamental domain with Im(z) A. Theorem. (RWC for the regularized fourth moment of Eisenstein series). Suppose that (.) holds for p = 4 and Ω = X, and p = 4 and Ω = D A for some A = A(T ) which tends to as T. Then we have reg E(z, + it ) 4 dµ(z) 7 π log T. X We have already explained above why (.) should be expected for p = 4 and Ω = X, even though for general p we must restrict to compact sets. The other possibility Ω = D A is already included in the RWC when A is fixed. But it is reasonable to conjecture that some effective error term will exist in (.), so that taking A which grows arbitrarily slowly should be permissible. We thank Matthew Young for suggesting this approach to us.

3 A CONJECTURE FOR THE REGULARIZED FOURTH MOMENT OF EISENSTEIN SERIES 3 Both of our main results are based on careful calculations arising from the regularized inner product. The point is to offer a new viewpoint for the fourth moment and carefully put into place all leading constants, so that the relevant conjecture might be verified in the future using the theory of L-functions.. Eisenstein series We recall the definition of Eisenstein series E(z, s) = Im(γz) s = ys γ Γ \Γ c,d Z (c,d)=, z H, cz + d s where Γ is the stabilizer of the cusp in Γ. The series is absolutely convergent in the halfplane Re(s) > where it defines an automorphic function satisfying E(z, s) = s( s)e(z, s), for the hyperbolic Laplacian = y ( x + y ). The Eisenstein series can be meromorphically continued to the whole s-plane and E(z, s) has the following Fourier expansion (for s,, ) E (z, s) = y s + ϕ(s)y s + (s) τ s / ( n ) yk s / (π n y)e(nx). n Here for complex α, τ α (n) = ab=n (a/b)α is the generalized divisor sum and the scattering function ϕ(s) can be explicitly expressed as (s ) ( s ϕ(s) =, where (s) = π s/ Γ ζ(s). (s) ) We will denote with e(y, s) := y s + ϕ(s)y s the constant term of the Eisenstein series. We denote with D = {z H z, x } the standard fundamental domain for and recall that its volume with respect to dµ is vol(x) = vol(d) = π 3. For a parameter A > we denote with D A := {z D Im(z) A} the corresponding truncated domain and with = D D A the corresponding cuspidal region. The truncated Eisenstein series { E(z, s), z DA E A (z, s) = E(z, s) e(y, s), z is now rapidly decreasing in the cusp. Calculation of the L -norm of this truncated Eisenstein series is done in [Sp, Section.3, both in the case of the whole fundamental domain and in the case of the cuspidal region, as follows: E A (z, + it ) dµz = ϕ ϕ ( + it ) + log A + AiT ϕ( it ) A it ϕ( + it ) it (.) D = log T + log A + O((log T ) /3+ɛ ) E A (z, + it ) dµz = 6 Aπ log T + O(A (log T ) /3+ɛ ) + O(A log A), T. Therefore in the compact truncated domain D A, since E = E A on D A, we have as T E(z, ( π D A + it ) dµz 3 ) 6 log T = vol(d A ) log T A π vol(x), as long as < A log T say. In other words, if we normalize the Eisenstein series as Ẽ(z, + it ) := E(z, + it ) log T,

4 4 GORAN DJANKOVIĆ AND RIZWANUR KHAN we have (.) lim T vol(d A ) Ẽ(z, D A + it ) dµz = vol(x). If we denote e iθ(t ) := (+it ) (+it ), then the function eiθ(t ) Ẽ(z, + it ) is real-valued, and the Random Wave Conjecture, as extended in [HR, predicts that e iθ(t ) Ẽ(z, +it ) tends to Gaussian N (, vol(x) / ) in distribution, when restricted to any compact and sufficiently regular subset Ω X. In particular, for the fourth moment (c 4 = 3), the conjecture predicts lim T vol(ω) Ω Ẽ(z, + it ) 4 dµz = 3 vol(x). By heuristic considerations and numerical experiments in [HR, the same limits should hold also for the normalized truncated Eisenstein series Ẽ A (z, + it ) := E A(z, + it ) log T. As explained, this should also include the case Ω = X, in which case the conjecture is (.3) E A (z, + it ) 4 dµz 36 π log T, as T. X 3. Regularized inner product and regularized Plancherel formula We will make use of the regularization process given by Zagier in [Za. An adelic version with a representation theoretic interpretation and with an alternate way of defining regularization is recently given in [MV. Let F (z) be a continuous Γ-invariant function on H. It is called renormalizable (in Zagier s terminology, or of controlled increase in the terminology of [MV) if there is a function Φ(y) on R > of the form (3.) Φ(y) = with c j, α j C and n j Z, such that l j= c j n j! yαj log nj y, F (z) = Φ(y) + O(y N ) as y, and for any N >. If F (z) = n= a n(y)e(nx) is the Fourier expansion of F at the cusp, in particular if a (y) is its -term, and if no α j equals or, then the function R(F, s) := (a (y) Φ(y))y s dy, where the defining integral converges for sufficiently large Re(s), can be meromorphically continued to all s and has a simple pole at s =. Then one can define the regularized integral with (3.) reg F (z)dµ(z) := π 3 Res s=r(f, s). Moreover, then the function F (z)e(z, s) with s, is also renormalizable and in particular, it can be shown that reg F (z)e(z, s)dµ(z) = R(F, s).

5 A CONJECTURE FOR THE REGULARIZED FOURTH MOMENT OF EISENSTEIN SERIES 5 It can be shown (see [Za) that the regularized integral can be written also as reg (3.3) F (z)dµ(z) = F (z)dµ(z) + D A (F (z) Φ(y))dµ(z) ˆΦ(A), where the right-hand side is independent of the value of the parameter A > and ˆΦ(y) is in the case α j for all j, given by the following explicit expression ˆΦ(y) = l j= c j y αj α j n j m= log m y m!( α j ) nj m. Under the assumption that no α j =, let E Φ (z) denote a linear combination of Eisenstein series E(z, α j ) (or suitable derivatives thereof) corresponding to all the exponents in (3.) with Re(α j ) > /, i.e. such that F (z) E Φ (z) = O(y / ). Then the third, equivalent definition of regularization is given by reg (3.4) F (z)dµ(z) = (F (z) E Φ (z))dµ(z). (3.5) For example Zagier showed in [Za that for s, s C \ {, }, s s, s, we have reg E(z, s )E(z, s )dµ(z) =. On the other hand, for the regularized product of the three Eisenstein series, Zagier (ibid. pg 43) obtained (3.6) reg E(z, + s )E(z, + s )E(z, + s 3)dµ(z) = = ( + s + s + s 3 )( + s s + s 3 )( + s + s s 3 )( + s s s 3 ). ( + s )( + s )( + s 3 ) The right-hand side is of course symmetric in s, s, s 3 because of the functional equation ( s) = (s). Since we are interested in the regularized product of 4 Eisenstein series, one can try to apply the definition (3.) directly. But already Zagier in [Za, pg. 43, discussed that in this case there is no useful closed-form expression for the result, as is for the product of 3 Eisenstein series in (3.6). Therefore, we must proceed indirectly via a regularized Plancherel formula. Now, let G(z) be another renormalizable Γ-invariant function such that G(z) = Ψ(y)+O(y N ) as y for any N >, where Ψ(y) = l d k k= m k! yβ k log m k y with d k, β k C. Then the product F (z)g(z) is also a renormalizable Γ-invariant function and if α j + β k, for all α j and β k appearing in Φ and Ψ respectively, the regularized inner product of F and G can be defined as reg F, G reg := F (z)g(z)dµ(z) = (F (z)g(z) E ΦΨ (z))dµ(z). It is easy to see from (3.3) that this regularized product is a Hermitian form. The regularized Plancherel formula from [MV is much more general, but for our purposes we will state and derive it entirely in classical situation of Zagier s paper [Za, much in the spirit of Lemma 4. from [Yo. Because of the cumbersome formulas, we will use the shorthand notation E s (z) := E(z, + s), and remind the reader not to confuse this with the truncated Eisenstein series E A (z, s) which still have arguments. Proposition 3. ([MV). Let F (z) and G(z) be renormalizable functions on such that F Φ and G Ψ are of rapid decay as y, for some Φ(y) = l c j j= n j! yαj log nj y and Ψ(y) =

6 6 GORAN DJANKOVIĆ AND RIZWANUR KHAN l k= d k m k! yβ k log m k y. Moreover, let α j, β k, Re(α j ), Re(β k), α j + β k and α j β k, for all j, k. Then the following formula holds: = F, 3/π reg 3/π, G reg + j F (z), G(z) reg = F, u j u j, G + 4π + F, E Ψ reg + E Φ, G reg. F, E it reg E it, G reg dt Proof. Because of the assumption Re(α j ), Re(β k), there exists some δ > such that F (z) := F (z) E Φ (z) = O(y / δ ) and G (z) := G(z) E Ψ (z) = O(y / δ ). For these F (z) and G (z) we have F (z)g (z) L () and hence F, G reg = F, G (the usual Petersson inner product), while also F (z), G (z) L () and hence one can apply the usual Plancherel formula for F (z), G (z), obtaining F (z), G(z) reg = F (z) + E Φ (z), G (z) + E Ψ (z) reg = = F, 3/π 3/π, G + j F, u j u j, G + 4π + F, E Ψ reg + E Φ, G reg + E Φ, E Ψ reg. F, E it E it, G dt Under our restrictions on the parameters α j, β k, all the inner products on the right hand side are well-defined and moreover because of (3.5), we have that E Φ, E Ψ reg =, which also implies that F, E Ψ reg = F, E Ψ reg and E Φ, G reg = E Φ, G reg. Furthermore, for the products with cusp forms u j we have F, u j = F, u j E Φ, u j = F, u j, the product with constant function is F, = F E Φ, = F, reg by definition of regularization and F, E it = F E Φ, E it reg = F, E it reg since E Φ, E it reg =, by (3.5). This finishes the proof. 4. Proof of Theorem. We want to apply this formula for the product of four Eisenstein series. By calculating the constant term of F (z) := E(z, + s )E(z, + s ), we find that E Φ (z) = E(z, + s + s ) + c E(z, s + s ) + c E(z, + s s ) + c c E(z, s s ) where ( ) c j = ϕ + s j = (s j) ( + s j ), and we have the similar formula for E Ψ (z) corresponding to G(z) := E(z, + s 3)E(z, + s 4). Hence, under the conditions on the parameters s j, j 4, described in Proposition 3. (α j = ± s ± s, β k = ± s 3 ± s 4 ), we get (4.) E s E s, E s3 E s4 reg = E(z, + s )E(z, + s ), E(z, + s 3)E(z, + s 4) = = 3 π E s E s, reg, E s3 E s4 reg + j E s E s, u j u j, E s3 E s4 + 4π E s E s, E it reg E it, E s3 E s4 reg dt + E s E s, E + c 3E +s3+s4 + c 4E s3+s4 + c 3c +s3 s4 4 E s3 s4 reg + E + c E +s+s + c E s+s + c c +s s E s s, E s 3 E s4 reg. For the cusp forms u j, the triple products E s E s, u j can be evaluated by the standard unfolding argument (see Section of [LS): reg

7 A CONJECTURE FOR THE REGULARIZED FOURTH MOMENT OF EISENSTEIN SERIES 7 Lemma 4.. Let u j (z) be a Hecke-Maass cusp form for the group Γ, that is an eigenvalue of the Laplace operator u j = ( 4 + t j )u j and of all Hecke operators T n u j = λ j (n)u j, for all n, which satisfies also T u j (z) = u j ( z) = ɛ j u j (z), ɛ j = ±. Then it has the Fourier expansion u j (z) = ρ j () n λ j(n) yk itj (π n y)e(nx) with λ j ( n) = ɛ j λ j (n). Let L(s, u j ) be the L-function associated to u j, defined by analytic continuation from the Dirichlet series n λ j(n)n s. Then for s, s ±/, if u j is even (ɛ j = ) we have (4.) E(, / + s )E(, / + s ), u j = ρ j() Λ( + s + s, u j )Λ( + s s, u j ), ( + s )( + s ) where Λ(s, u j ) := π s Γ( s+itj )Γ( s itj )L(s, u j ) is the completed L-function corresponding to u j. In the case of odd u j (i.e. ɛ j = ), the triple product is. Remark: The right hand side in (4.) is symmetric in s, s, since for even u j, we have the functional equation Λ(s, u j ) = Λ( s, u j ). Moreover, we have the following formula relating the normalizing factor ρ j () with the symmetric square L-function: ρ j () = cosh(πt j) L(, sym u j ). Further, for s ±s and s 3 ±s 4, by (3.5) the first term on the right-hand side in (4.) vanishes. Finally, using (3.6) for all regularized triple products of Eisenstein series, we arrive at (4.3) E s E s, E s3 E s4 reg = j ɛ j= + cosh(πt j ) Λ( + s + s, u j )Λ( + s s, u j )Λ( + s 3 + s 4, u j )Λ( + s 3 s 4, u j ) L(, sym u j )( + s )( + s )( + s 3 )( + s 4 ) + 4π δ (,δ {±} + ti + δ s + δ s )( + ti + δ s 3 + δ s 4 ) ( + ti) dt ( + s )( + s )( + s 3 )( + s 4 ) δ,δ {±} ( + δ s + δ s + s 3 + s 4 ) ( + s )( + s )( + s 3 + s 4 ) δ +c ( + δ,δ {±} s + δ s + s 3 s 4 ) 4 ( + s )( + s )( + s 3 s 4 ) + δ,δ {±} ( + s + s + δ s 3 + δ s 4 ) ( + s + s )( + s 3 )( + s 4 ) δ +c ( + s,δ {±} s + δ s 3 + δ s 4 ) ( + s s )( + s 3 )( + s 4 ) + c 3 δ,δ {±} ( + δ s + δ s s 3 + s 4 ) ( + s )( + s )( s 3 + s 4 ) + c 3 c 4 δ,δ {±} ( + δ s + δ s s 3 s 4 ) ( + s )( + s )( s 3 s 4 ) δ + c ( s,δ {±} + s + δ s 3 + δ s 4 ) ( s + s )( + s 3 )( + s 4 ) δ + c c ( s,δ {±} s + δ s 3 + δ s 4 ) ( s s )( + s 3 )( + s 4 ) Let us denote the last eight terms (quotients of products of -functions, coming from the regularization process) on the right hand side of (4.3) with Ξ j, j 8, respectively with the order of appearance in (4.3). Now, let us choose for s j the following values: s = it, s = it + ν, s 3 = it and s 4 = it + η, with complex parameters ν and η satisfying < Re(ν) < Re(η) < 4. For these values all the conditions from Proposition 3. are satisfied. From (3.4), we see that E it E it +ν, E it E it +η reg is continuous in ν, η, and therefore, if we first let ν in (4.3), keeping η fixed, we get (4.4) E it, E it E it +η reg = j + 4π ɛ j= cosh(πt j ) Λ( + T i, u j)λ( T i + η, u j)λ(, u j)λ( η, u j) L(, sym u j ) ( + T i)( T i)( T i + η) ( + ti) ± ( + ti ± T i)( + ti ± T i η)( + ti ± η) ( + ti) dt ( + T i)( T i)( T i + η).

8 8 GORAN DJANKOVIĆ AND RIZWANUR KHAN where + 8 Ξ j (T, η), j= Ξ (T, η) = ( + η) ( T i + η)( 4T i + η) ( + T i)( 4T i + η) Ξ (T, η) = ( + η)( + T i + η)( T i + η) ( T i)( + T i)( + η) =: ( + η)f (η), =: ( + η)f (η), Ξ 3 (T, η) = ( η)( + T i η)( + T i η)( T i η) ( + T i)( T i + η)( η) =: ( η)f 3 (η), Ξ 4 (T, η) = ( η) ( + T i η)( + T i η)( + 4T i η) ( + T i)( T i)( T i + η)( + 4T i η) =: ( η)f 4(η), ( + η)( + T i η)( + T i + η)( + 4T i η) Ξ 5 (T, η) = =: ( + η)f 5 (η), ( + 4T i)( T i)( T i + η) ( η)( + η)( + T i η)( T i + η) Ξ 6 (T, η) = Ξ 7 (T, η) = =: ( η)( + η)f 6 (η), ()( + T i)( T i + η) ( η)( T i η)( T i)( T i + η)( 4T i + η) Ξ 8 (T, η) = =: ( η)f 8 (η). ( + T i)( T i + η)( 4T i) Each of Ξ j has a pole at η =, but the whole sum 8 j= Ξ j has a removable singularity at η =. This can be seen by grouping together Ξ with Ξ 8, Ξ 4 with Ξ 5, and Ξ + Ξ 3 with Ξ 6 = Ξ 6 + Ξ 7. More explicitly, if we denote with (4.5) (s) = s + a + b(s ) + O((s ) ) the Laurent expansion of (s) around s =, we get the following expansions of Ξ j (η): Ξ (η) = F () η + F () + af () +O(η), Ξ (η) = F () η + F () + af () + (a + b)f () + af η () + F () +O(η), Ξ 3 (η) = F 3() η + F 3() af 3 () + (a + b)f 3 () af η 3() + F 3 () +O(η), Ξ 4 (η) = F 4() F η 4() + af 4 () +O(η), Ξ 5 (η) = F 5 () + F η 5() + af 5 () +O(η), Ξ 6 (η) = Ξ 7 (η) = F 6() η F 6() + (a b)f 6 () η F 6 () +O(η), Ξ 8 (η) = F 8() F η 8() + af 8 () +O(η). But, F () = F 3 () = F 6 () = (), so the polar terms with η cancel out in the sum. Further, F () = F 8 (), F 4 () = F 5 () and one calculates and F () = () [Re ( + T i) (), F 3() = () F 6() = () Re ( + T i), [ () 3Re ( + T i)

9 A CONJECTURE FOR THE REGULARIZED FOURTH MOMENT OF EISENSTEIN SERIES 9 from which it follows that F () + F 3() F 6() =, a.e. the coefficient in front of η also vanishes. Therefore we can take η in (4.4) and after calculation of all the other required derivatives appearing in 8 lim Ξ j (T, η) = a(f () + F 4 () + F 5 () + F 8 ()) + a (F () + F 3 () + F 6 ()) η j= +F () F 4() + F 5() F 8() + a(f () F 3()) + F () + F 3 () F 6 (), we obtain the following exact evaluation of the regularized fourth power of Eisenstein series: Proposition 4.. For any nonzero real T, we have: (4.6) = j ɛ j= + 4π [ + 4 () reg E(z, / + it ) 4 dµ(z) = E it, E it reg cosh(πt j ) Λ( + T i, u j)λ( T i, u j)λ (, u j) L(, sym u j ) ( + T i) 4 Re 4 ( + ti) ( + ti + T i) ( + ti T i) ( + ti) ( + T i) 4 dt ( + T i) + ( + T i) + Re ( ) +4(a ())Re ( + T i) + ) ( + ( + T i)( + 4T i) [a + 4 ( T i)( + 4T i) [a ( T i)( 4T i) ( + T i)( 4T i) where a = lim s ((s) (s ) ). () ( + T i) () a () + a ( + T i) ( + 4T i) ( T i) ( 4T i) Remark: The exact value of the constant a is C ln π ln = , where C = is Euler s constant. This is a consequence of the following two formulas: ζ(s) = )) and Γ Γ (/) = C ln. Also we recall that () = π 6. Using Stirling s approximations Γ(σ + it) = e π t / t σ, s +C +O((s π{ + O( t )} and Γ Γ (s) = log s + O( s ) valid in a fixed vertical strip when t, and classical estimates for the Riemann zeta-function on the edge of the critical strip [MoV, section 6.3, (4.7) (log t) /3 (log log t) /3 ζ(s) (log t) /3 and for s = σ + it, σ (log t) /3 (log log t) /3, ζ ζ (s) (log t)/3 (log log t) /3 one obtains first that ( ± T i) log T and then that the contribution of the terms in the last two lines in the formula (4.6) is O( log T ). Therefore the contribution on the right-hand side of T / (4.6) coming from the regularization process is 4 π [ Re ( + T i) + ( + T i) + Re ( ) ( + T i) + O(log T ).

10 GORAN DJANKOVIĆ AND RIZWANUR KHAN Since log π (s) = + Γ Γ ( s ) + ζ ζ (s), from Stirling s approximation and (4.7) we obtain further that when T ( + T i) + Re ( ) ( + T i) = 3 4 log T + O(log 5/3+ɛ T ), for any ɛ >. Lemma 4.3. As t, we have ζ ζ ( + ti) log4/3+ɛ t. Proof. We can use (4.7) and the Borel-Carathéodory lemma [MoV, Lemma 6. to get the bound ( ) ζ ( + ti) log 4/3+ɛ t. ζ This and (4.7) again imply the stated bound for ζ ζ ζ ( + ti) = ( ζ ) ( + ti) + ( ζ ζ ( + ti)). From (s) = log π + Γ 4 4 Γ ( s ) + ζ ζ (s) log π Γ Γ ( s Γ ) (log π)ζ (s) + ζ Γ ( s )ζ ζ (s), using (4.7), Lemma 4.3 and another well-known approximation Γ Γ ( + T i) = log T + O(log T ), we obtain also the asymptotic Re ( + T i) = 4 log T + O(log 5/3+ɛ T ). On the other hand, the contribution of the continuous spectrum in (4.6) i.e. the integral on the right-hand side is of a smaller size, being bounded by the integral in the following Lemma: Lemma 4.4. For T we have: for some absolute implicit constant. ( + ti) 4 ( + ti + T i) ( + ti T i) ( + ti) ( + T i) 4 dt T /6, Proof. This is exactly Proposition 3.4 in [Sp. For completeness we briefly repeat here the argument. After employing Stirling s asymptotic formula for Gamma functions and after splitting the integral + = 3T + + 3T, one can see easily that the contribution in the range + 3T decays exponentially with T. Therefore, one needs to bound the integral 3T ζ( + ti) 4 ( + t ) ζ( + (t + T )i) ζ( + (t T )i) ( + t + T ) / ( + t T ) / e π (4T t T t+t ) ζ( + ti) ζ( + T i) 4 dt. By (4.7) the fourth ratio can be bounded by T ε, the third ratio can be bounded by convexity bound, while the second ratio can be bounded using the subconvexity bound ζ( + ti) ( + t ) θ+ε for some θ < 6, which is available and sufficient. The bound follows by the fourth moment estimate 3T ζ( +ti) 4 + t dt T ε, for any ε >. Therefore after putting together everything in this section, we obtain the asymptotic formula in Theorem.. Note that the theorem has dropped the condition ɛ j =. This is fine because when ɛ j =, we have Λ(, u j) = and the summand vanishes.

11 A CONJECTURE FOR THE REGULARIZED FOURTH MOMENT OF EISENSTEIN SERIES 5. Proof of Theorem. The regularized fourth moment of Eisenstein series E(z, / + it ) E it, E it reg = reg E(z, + it ) 4 dµ(z) can also be expressed directly using (3.3). The corresponding function Φ(y) is given by Φ(y) = e(y, / + it ) 4 = c y 4T i + 4cy T i + 6y + 4cy +T i + c y +4T i where c = ϕ(/ + it ) = ( T i) (+T i), so in particular c =. Hence we get reg E(z, + it ) 4 dµ(z) = = E(z, / + it ) 4 dµ(z) + D A ( E(z, / + it ) 4 e(y, / + it ) 4 )dµ(z) ˆΦ(A), with ˆΦ(A) given explicitly by ˆΦ(A) = c A 4T i 4T i + 4c A T i T i + 6A + 4c A+T i + T i + c A+4T i + 4T i. In particular, ˆΦ(A) A. This will be an admissible error for all the values of the truncation parameter in the range < A log T. Hence the difference between the regularized integral of E 4 and the integral X E A 4 dµ with the truncated Eisenstein series considered in [Sp is reg E(z, + it ) 4 dµ(z) E A (, + it ) 4 4 = E 4 dµ + D A ( E A + e 4 e 4 )dµ ˆΦ(A) E 4 dµ D A E A 4 dµ (5.) = (e E A + e EA + 4 e E A )dµ + ( E A E A e + E A E A e)dµ ˆΦ(A), since E A e e dµ = E A e e dµ =. Here, the first integral in the cuspidal region can be explicitly computed. From the integral representation K ν (y) = + e y cosh t cosh(νt)dt, Re(ν) >, we see that K it (y) is real for y >, T R and hence for z (5.) ( + T i)e A (z, / + it ) = 4 τ it (n) y KiT (πny) cos(πnx) n= is also real-valued. Using this and the functional equation for (s), after a short calculation one gets that the first integral in (5.) is equal to ( ) y ( + T i) + 6y+T i ( T i) + 6y T i ( + T i)e ( + T i) A(z, / + it )dµ(z). Therefore we need to calculate the twisted integrals of the second moment of the truncated Eisenstein series in the cuspidal region I η := y +η ( + T i)ea(z, / + it )dµ(z),

12 GORAN DJANKOVIĆ AND RIZWANUR KHAN for the values of parameter η {, ±T i}. Substituting here the Fourier expansion (5.) we obtain + ( ) I η = 6 y +η τ it (n) y dxdy KiT (πny) cos(πnx) where = 8 A τit (n) n= A n= K it (πny)y η dy = 8 g(x) := The Mellin transform of this function is equal to x y τit (n)(πn) η g(πan), n= K it (y)y η dy. G(s) := g(x)x s dx x = KiT (x)x η+s dx s η +s ( ) ( ) ( ) + η + s + η + s + η + s = sγ( + η + s) Γ Γ + it Γ it by integration by parts and the Mellin-Barnes formula [GR, K µ (y)k ν (y)y s dy y = s 3 ( ) s ± µ ± ν Γ. Γ(s) ±,± By the inverse Mellin transform we have g(x) = πi (3) G(s)x s ds (where the integration is over the line Re(s) = 3) and so we get 8 I η = (π) +η G(s)(πA) s τit (n) ds. πi ns++η (3) Here, since τ it (n) = σ it (n)n it, we have by Ramanujan s identity τit (n) n s++η = ζ (s + + η)ζ(s + + η + T i)ζ(s + + η T i), ζ(s + + η) which then gives n= I η = A s πi (4) s n= (s + η)(s + η + T i)(s + η T i) ds. (s + η) The integrand is rapidly decreasing in vertical strips and it is regular on the line Re(s) = (for all three values of the parameter η), so we can shift the line of integration from Re(s) = 4 to Re(s) = : (5.3) I η = R η + πi (/) A s s (s + η)(s + η + T i)(s + η T i) ds, (s + η) where R η = P R η,p is the sum of residues R η,p of the poles P that we encounter. In the case η =, the integrand has two simple poles at s = ± T i with the residues R, T i = AT i ( T i)( 4T i) T i ( 4T i) and the triple pole at s = with residue [ ( + T i) R, = () ( + T i) + Re ( (log A)Re ( + T i) + () a and R,+T i = A T i ( + T i)( + 4T i) T i ( + 4T i) ( + T i) + log A ) (log A Re ) ( + T i)

13 A CONJECTURE FOR THE REGULARIZED FOURTH MOMENT OF EISENSTEIN SERIES 3 where the constants a and b are as in (4.5). ( ) +4 () 4a () + a + b (), In the case η = T i, the integrand has the simple pole at s = 4T i with the residue R T i, 4T i = A4T i ( T i)( 4T i) 4T i ( 4T i) and two double poles at s = and s = T i with the corresponding residues and R T i, = ( + T i)( + 4T i) ( + 4T i) [a log A + R T i, T i = AT i ( + T i)( T i) T i () ( + 4T i) + ( + T i) ( + 4T i) [Re ( + T i) log A + + a T i (). In the case η = T i, the integrand has the simple pole at s = + 4T i with the residue R T i,+4t i = A 4T i ( + T i)( + 4T i) 4T i ( + 4T i) and two double poles at s = and s = + T i with the corresponding residues and R T i, = ( T i)( 4T i) ( 4T i) [a log A + R T i,+t i = A T i ( + T i)( T i) T i () In particular, we have when T. ( 4T i) + ( T i) ( 4T i) [Re ( + T i) log A + a T i (). ( ± T i) R T i (log T ) T /, The contribution of the integrals on the shifted line in (5.3) is bounded in the following Lemma: Lemma 5.. For any η {, ±T i} with T >, we have A ti ti ( + ti + η)( + ti + η + T i)( + ti + η T i) ( + T i)( + ti + η) dt A/ T /6, with an absolute implicit constant. Proof. The analysis is similar to that in Lemma 4.4. The case η = was treated in [Sp, section 4.3., where the bound O(A / T /6 ) is obtained. Here, we treat the case η = T i (for η = T i, the value of the integral is the same). Using Stirling s formula, we see that the integrand is bounded by A / e π 4 (4T t t+4t ) ( + t ) 5/4 ( + t + T ) / ( + t + 4T ) /4 ζ( + ti) ζ ( + (t + T )i) ζ( + (t + 4T )i) ζ ( + T i) ζ( + (t + 4T )i).

14 4 GORAN DJANKOVIĆ AND RIZWANUR KHAN Using subconvexity estimate ζ(/ + ti) ( + t ) θ+ɛ, for all ɛ > and some θ < 6 for the zetafunctions in the numerator and (4.7) for the zeta-functions in the denominator, this is further bounded by A / T ɛ e π 4 (4T t t+4t ) ( + t ) 5 4 θ ɛ ( + t + T ) θ ɛ ( + t + 4T ). 4 θ ɛ We split the integration into 3 ranges: = 4T + 4T +. In the first and the third range we have an exponential decay of the integrand and so we have that in these ranges the integrals are bounded respectively by A / T ɛ e π t ( + t ) 5 4 θ ɛ T θ+ 4 θ dt A/ T /4 and 4T 4T A / T ɛ e π (t+4t ) T 5 4 θ+ θ ( + t + 4T ) dt 4 θ ɛ A/ T 5/4. In the middle range the integral is bounded by A / T ɛ A / T θ 4 +ɛ T 4T dt ( + t ) 5 4 θ ( + t + T ) θ ( + t + 4T ) 4 θ dt ( + t ) 5 4 θ ( + t + T ) θ A/ T 3θ 3 4 +ɛ A / T /4. Therefore, in the cases η = ±T i, we get an even better bound O(A / T /4 ). After we collect everything together, and use asymptotic formulas for ( + it ) and ( + it ) already seen in the previous section, we get that the contribution of the first integral in (5.) is = ( + T i) I + ( + T i) R + 6 ( T i) I T i + 6 ( T i) R T i + 6 ( + T i) I T i 6 ( + T i) R T i + O(A / T /6 ) (5.4) = 36 π log T + O(log 5 3 +ɛ T ), for the range < A log T. The main contribution is coming from R,. Putting together all our calculations in this section, we end up with the following Proposition: Proposition 5.. When T, for any value of the truncation parameter < A log T we have (5.5) X reg E A (z, + it ) 4 dµ(z) + 36 π log T + X E(z, + it ) 4 dµ(z) = ( E A E A e + E A E A e)dµ + O(log 5 3 +ɛ T ). The first integral on the right hand side of (5.5) is asymptotic to 36 π log T under assumption (.3). The integral over cuspidal region in (5.5) is bounded by (5.6) 4 E3 A(z, + T i)e(y, ( ) / ( ) / + T i) dµ(z) 4 E A 4 dµ ee A dµ.

15 A CONJECTURE FOR THE REGULARIZED FOURTH MOMENT OF EISENSTEIN SERIES 5 The second integral on the right hand side of (5.6) is 6 π log T ; this is implicit in the calculation of the first integral in (5.). Under the RWC, the first integral on the right hand side of (5.6) can be bounded by E A 4 dµ E A 4 dµ (vol(x) vol(d A )) log T A log T, X D A which is o(log T ) if A grows arbitrary slowly to infinity as T. This way the right hand side of (5.5) is asymptotic to 7 π log T. Acknowledgment. Work on this project started when the first author visited Texas A&M University at Qatar. He wishes to thank that institution for hospitality and excellent working conditions. References [BK J. Buttcane, R. Khan, On the Fourth Moment of Hecke Maass Forms and the Random Wave Conjecture, Compos. Math., to appear. [GR I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series and Products, Seventh Edition, Academic Press, 7 [HR D. Hejhal, B. Rackner, On the Topography of Maass Waveforms for P SL(, Z), Experimental Mathematics, Vol. (99), No. 4, 75-35, 4 [LS W. Luo, P. Sarnak, Quantum ergodicity of Eigenfunctions on P SL (Z) \ H, Inst. Hautes Études Sci. Publ. Math. No. 8 (995), 7-37, 6 [Li E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. Of Math. (), Vol. 63 (6), No., [Lu W. Luo, L 4 -norm of the dihedral Maass forms, Int. Math. Res. Notices (4) 4 (8): [MV P. Michel, A. Venkatesh, The subconvexity problem for GL, Publ. Math. Inst. Hautes Études Sci. () 7-7 4, 5 [MoV H.L. Montgomery, R. C. Vaughan, Multiplicative Number Theory: I. Classical Theory, Cambridge University Press, 6 9, [So K. Soundararajan, Quantum unique ergodicity for SL (Z)\H, Ann. of Math. (), Vol. 7 (), No., [Sp F. Spinu, The L 4 Norm of the Eisenstein Series, PhD Dissertation, Princeton University, 3,, 3,,, 3 [Yo M. Young, The quantum unique ergodicity conjecture for thin sets, Advances in Mathematics 86(6), [Za D. Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J.Fac. Sci. Univ. Tokyo Sect. IA Math. 8(3) (98) , 4, 5 University of Belgrade, Faculty of Mathematics, Studentski Trg 6, p.p. Serbia address: djankovic@matf.bg.ac.rs 55, Belgrade, Texas A&M University at Qatar, PO Box 3874, Doha, Qatar address: rizwanur.khan@qatar.tamu.edu