The adjoint map of the Serre derivative and special values of shifted Dirichlet series
aa r X i v : . [ m a t h . N T ] F e b THE ADJOINT MAP OF THE SERRE DERIVATIVE AND SPECIALVALUES OF SHIFTED DIRICHLET SERIES
ARVIND KUMAR
Abstract.
We compute the adjoint of the Serre derivative map with respect to the Pe-tersson scalar product by using existing tools of nearly holomorphic modular forms. TheFourier coefficients of a cusp form of integer weight k , constructed using this method, in-volve special values of certain shifted Dirichlet series associated with a given cusp form f of weight k + 2. As application, we get an asymptotic bound for the special values of theseshifted Dirichlet series and also relate these special values with the Fourier coefficients of f .We also give a formula for the Ramanujan tau function in terms of the special values of theshifted Dirichlet series associated to the Ramanujan delta function. Introduction
For any positive integer k ≥
4, let M k (Γ) (resp. S k (Γ)) be the space of modular forms(resp. cusp forms) of weight k for a congruence subgroup Γ of SL ( Z ). For even k ≥
2, theEisenstein series of weight k is given by E k ( z ) = 1 − kB k ∞ X n =1 σ k − ( n ) q n , where B k is the k th Bernoulli number, σ k − ( n ) = P d | n d k − , q = e πiz , and z is in the upperhalf-plane H . For k ≥ E k is a modular form of weight k for SL ( Z ). The Eisenstein series E and the derivative Df := 12 πi ddz of f ∈ M k (Γ) do notsatisfy the modular property. But by taking a certain linear combination of Df and E f ,we get a function which transforms like a modular form of weight k + 2. The underlyingmap on M k (Γ) is called as the Serre derivative, denoted by ϑ k and given in section 3. Wefirst show that the Serre derivative is a C -linear map from S k (Γ) to S k +2 (Γ) which are finitedimensional Hilbert spaces. In this article our purpose is to find the adjoint of the Serrederivative map with respect to the Petersson inner product. It gives construction of a cuspform of weight k with interesting Fourier coefficients, from a given cusp form of weight k + 2.Using the properties of Poincar´e series and adjoint of linear maps, W. Kohnen in [14]constructed the adjoint map of the product map by a fixed cusp form, with respect to thePetersson scalar product. After Kohnen’s work, similar results in various other spaces have Date : May 10, 2018.2010
Mathematics Subject Classification.
Primary 11F25, 11F37; Secondary 11F30, 11F67.
Key words and phrases.
Modular forms, Nearly holomorphic modular forms, Serre derivative, Adjointmap. been obtained by many mathematicians, since the Fourier coefficients of the image of aform involve special values of certain shifted Dirichlet series attached to these forms, e.g.,generalization to Jacobi forms (see [5], [10] and [18]), Siegel modular forms (see [15] and[11]), Hilbert modular forms (see [21]) and half-integral weight modular forms (see [9]).In this case also, the Fourier coefficients of the image of f under the adjoint of the Serrederivative map, involve special values of certain shifted Dirichlet series associated with theFourier coefficients of f . As an application we obtain relations among the coefficients ofmodular forms and special values of these shifted Dirichlet series. Moreover, we also givean asymptotic bound of the special values of this shifted Dirichlet series. Since E and thederivative of a modular form are quasimodular forms (introduced by Kaneko and Zagier[12]), they do not satisfy the modular transformation property. Hence, it is not possible todefine the Petersson inner product in the usual way for the space of quasimodular forms.However, there is an isomorphism between the space of quasimodular forms and the spaceof nearly holomorphic modular forms and we can define the Petersson inner product in thespace of nearly holomorphic modular forms. Therefore, sometimes it is convenient to switchour problems from quasimodular forms to nearly holomorphic modular forms and vice versa.By the means of Maass-Shimura operator R k and E ∗ (see section 2 for notations), we firsttransform the definition of the Serre derivative in the context of nearly holomorphic modularforms and then we compute the adjoint map explicitly. We emphasize that our proof can becarried out in the setting of half-integral weight forms to construct cusp forms of half-integerweight. 2. Basic tools and notations
Let µ Γ denote the index of Γ in SL ( Z ). Unless otherwise stated we assume that k ∈ Z .For γ = (cid:16) a bc d (cid:17) ∈ GL +2 ( Q ), we define the slash operator as follows: f | k γ ( z ) := (det γ ) k/ ( cz + d ) − k f ( γz ) , where γz = az + bcz + d . We denote by M k (Γ , χ ) (resp. S k (Γ , χ )) the space of modular forms (resp. cusp forms) ofweight k for the congruence subgroup Γ of level N and Dirichlet character χ (mod N ). Wewrite M k ( N ) (resp. S k ( N )) for the corresponding spaces if Γ = Γ ( N ) and χ is the principalcharacter.Let f, g ∈ M k (Γ) be such that the product f g is a cusp form. Write z = x + iy , then thePetersson inner product is defined by h f, g i := 1 µ Γ Z z ∈ Γ \H f ( z ) g ( z ) y k dxdyy . (1) HE ADJOINT MAP OF THE SERRE DERIVATIVE AND SPECIAL VALUES OF SHIFTED DIRICHLET SERIES3
Let k and n be positive integers. The n th Poincar´e series of weight k for a congruencesubgroup Γ is defined by P k,n ( z ) := X γ ∈ Γ ∞ \ Γ e πinz | k γ, (2)where Γ ∞ := (cid:26) ± (cid:16) t (cid:17) | t ∈ Z (cid:27) . It is well known that P k,n ∈ S k (Γ) for k > Lemma 2.1.
Let f ∈ S k (Γ) with Fourier expansion f ( z ) = ∞ P m =1 a ( m ) q m . Then h f, P k,n i = α k,n a ( n ) , where, α k,n = Γ( k − πn ) k − . The following familiar result tells about the growth of Fourier coefficients of a modularform in which the first statement is trivial and the second is due to P. Deligne [6]:
Proposition 2.2. If f ∈ M k (Γ , χ ) with Fourier coefficients a ( n ) , then a ( n ) ≪ n k − ǫ , and moreover, if f is a cusp form, then a ( n ) ≪ n k − + ǫ , for any ǫ > . Nearly holomorphic modular forms.
For the convenience of the reader we repeatthe relevant material from [20] in our setting without proofs, thus making our expositionself-contained.
Definition 2.3.
A nearly holomorphic modular form f of weight k and depth ≤ p for Γ isa polynomial in ℑ ( z ) of degree ≤ p whose coefficients are holomorphic functions on H withmoderate growth such that f | k γ ( z ) = f ( z ) , for any γ = (cid:18) a bc d (cid:19) ∈ Γ and z ∈ H , where ℑ ( z ) is the imaginary part of z . Let c M ≤ pk (Γ) denote the space of nearly holomorphic modular forms of weight k and depth ≤ p for Γ. We denote by c M k (Γ) = S p c M ≤ pk (Γ) the space of all nearly holomorphic modularforms of weight k . Note that E ∗ ( z ) = E ( z ) − π ℑ ( z ) (3)is a nearly holomorphic modular form of weight 2 for the group SL ( Z ). Definition 2.4.
Let f ∈ c M k (Γ) . Then f is called a A. KUMAR • rapidly decreasing function at every cusp of Γ if for each α ∈ SL ( Q ) and positivereal number c , there exist positive constants A and B depending on f , α and c suchthat |ℑ ( αz ) k/ f ( αz ) | < Ay − c i f y = ℑ ( z ) > B. • slowly increasing function at every cusp of Γ if for each α ∈ SL ( Q ) there existpositive constants A , B and c depending on f and α such that |ℑ ( αz ) k/ f ( αz ) | < Ay c if y = ℑ ( z ) > B. Remark . For example, a modular form is slowly increasing and a cusp form is rapidlydecreasing function. Moreover, the product of a rapidly decreasing function with any nearlyholomorphic modular form provides a rapidly decreasing form.Let f, g ∈ c M k (Γ) be such that the product f g is a rapidly decreasing function. Write z = x + iy , then the inner product is defined by h f, g i := 1 µ Γ Z z ∈ Γ \H f ( z ) g ( z ) y k dxdyy . (4)By abuse of notation, we used the same symbol here for Petersson inner product as in thecase of modular forms given in (1). The integral is convergent because of the hypothesis andhence the inner product is well defined. Definition 2.5.
The Maass-Shimura operator R k on f ∈ c M k (Γ) is defined by R k ( f ) = 12 πi (cid:18) k i ℑ ( z ) + ∂∂z (cid:19) f ( z ) . The operator R k takes c M k (Γ) into c M k +2 (Γ), so sometimes it is called Maass raising operator.There is another operator L k := − y ∂∂z : c M k +2 (Γ) → c M k (Γ), known as Maass loweringoperator, which annihilates any holomorphic function. In [20, Theorem 6.8], It was shownthat the operator L k is the adjoint of R k with respect to the inner product (4), as long asthe product of the functions is rapidly decreasing (see also [2, Lemma 4.2]). We now statean interesting application of this observation which plays a crucial role in the proof of ourmain result. Lemma 2.6.
Let Γ be a congruence subgroup and let f ∈ S k +2 (Γ) . Then h f, R k g i = 0 forany g ∈ M k (Γ) . A shifted Dirichlet series.
Let f ( z ) = ∞ P n =0 a ( n ) q n and g ( z ) = ∞ P n =0 b ( n ) q n . For m > L f,g,m ( s ) = X n > a ( n + m ) b ( n )( n + m ) s . (5) HE ADJOINT MAP OF THE SERRE DERIVATIVE AND SPECIAL VALUES OF SHIFTED DIRICHLET SERIES5
If the coefficients a ( n ) and b ( n ) satisfy an appropriate bound, then L f,g,m ( s ) converges ab-solutely in some half-plane.For f ∈ S k (Γ , χ ) and a non negative integer m , consider a shifted Dirichlet series of Rankintype associated with f and E denoted by L f,m ( s ) and defined by L f,m ( s ) := − L f,E ,m ( s ) = X n > a ( n + m ) σ ( n )( n + m ) s . (6)Then by Proposition 2.2, it is absolutely convergent for Re( s ) > k +32 . It can be shownthat L f,m ( s ) has a meromorphic continuation to C (compare [8, section 2-5]). A slightlydifferent shifted Dirichlet series of this kind associated with two modular forms was firstintroduced by Selberg in [19]. Recently, in [8], J. Hoffstein and T. A. Hulse rigorouslyinvestigated the meromorphic continuation of a variant of Selberg’s shifted Dirichlet seriesand multiple shifted Dirichlet series. In [16], M. H. Mertens and K. Ono proved that certainspecial values of symmetrized sum of such functions involve as the coefficients of sum ofmixed mock modular forms and quasimodular forms. The p -adic properties of these werefurthermore studied in [1]. 3. Serre derivative
It is well known [17, Proposition 2.11] that for a positive integer k and f ∈ M k (Γ) thefunction ϑ k f := Df − k E f (7)is a modular form of weight k + 2 for Γ. The weight k operator ϑ k defined by (7) is calledthe Serre derivative (or sometimes the Ramanujan-Serre differential operator ). It is aninteresting and useful operator because it defines an operator on modular forms for anycongruence subgroup with character and also preserves cusp forms.
Theorem 3.1.
Let k be a non negative integer. Then the weight k -operator ϑ k maps M k (Γ , χ ) to M k +2 (Γ , χ ) and S k (Γ , χ ) to S k +2 (Γ , χ ) . In particular ϑ k maps S k ( N ) to S k +2 ( N ) .Remark . We observe that, the Serre derivative can also be written in the form ϑ k ( f ) = R k f − k E ∗ f. (8)This form is quite useful while computing the Petersson inner product as R k f and E ∗ f arenearly holomorphic modular forms, where the inner product is defined by (4), provided f isa cusp form. Remark . Similar to (7), we can define the weight k -operator ϑ k/ for an odd positiveinteger k . Then one can easily see that ϑ k/ maps a modular form (resp. cusp form) ofweight k to modular form (resp. cusp form) of weight k + 2. A. KUMAR Main Theorem
From Theorem 3.1 we know that ϑ k : S k (Γ) → S k +2 (Γ) is a C − linear map of finitedimensional Hilbert space and hence has an adjoint map ϑ ∗ k : S k +2 (Γ) → S k (Γ), such that h ϑ ∗ k f, g i = h f, ϑ k g i , ∀ f ∈ S k +2 (Γ) , g ∈ S k (Γ) . In the main result we exhibit the Fourier coefficients of ϑ ∗ k f for f ∈ S k +2 (Γ). Its m th Fouriercoefficient involves special values of the shifted Dirichlet series L f,m ( s ). Now we shall statethe main theorem of this article. Theorem 4.1.
Let k >
2. The image of any function f ( z ) = P n > a ( n ) q n ∈ S k +2 (Γ) under ϑ ∗ k is given by ϑ ∗ k f ( z ) = X m > c ( m ) q m , where c ( m ) = 1 µ Γ k ( k − m k − (4 π ) " ( m − k ) m k +1 a ( m ) + 2 kL f,m ( k + 1) . Applications
An asymptotic bound for L f,m ( k + 1) . Let f ∈ S k +2 (Γ) and Γ be a congruencesubgroup of level N . From Theorem 4.1, we can write L f,m ( k + 1) = 12 k " µ Γ (4 π ) k ( k −
1) 1 m k − c ( m ) − ( m − k ) m k +1 a ( m ) . Here, c ( m ) is the m th Fourier coefficient of ϑ ∗ k f which is a cusp form of weight k . Hence, inview of Proposition 2.2, a direct calculation gives L f,m ( k + 1) ≪ m − k , (9)where the implied constant depends on f .5.2. Values of L f,m ( k + 1) in terms of the Fourier coefficients. Let k > S k (Γ) is a one-dimensional space; we denote a generator of S k (Γ) by f ( z ). Then applying Theorem 4.1, we get ϑ ∗ k g ( z ) = α g f ( z ) for any g ∈ S k +2 (Γ),where α g is a constant. Now equating the m th Fourier coefficients both the sides, we get arelation among the special values of the shifted Dirichlet series associated with g and theFourier coefficients of f . In the following, we illustrate this with one example.From now on, ∆ k,N will denote the unique normalized cusp form with Fourier coefficients τ k,N ( n ) in the one dimensional space S k ( N ). Note that ∆ , ( z ) = ∆( z ), whose Fouriercoefficients τ ( n ), the Ramanujan tau function. For a positive integer t , we introduce the V -operator acting on a function f (defined on C ) by V t f ( z ) := f ( tz ) . HE ADJOINT MAP OF THE SERRE DERIVATIVE AND SPECIAL VALUES OF SHIFTED DIRICHLET SERIES7
It is known that V t is a linear operator from S k ( N ) into S k ( N t ). Note that S (2) = C ∆ , ( z )and S (2) = C ∆( z ) ⊕ C V ∆( z ). Now considering the map ϑ : S (2) → S (2), a directcomputation shows that ϑ ∆ , ( z ) = 16 ∆( z ) + 1283 V ∆( z ) (10)Let ϑ ∗ ∆( z ) = α ∆ , ( z ) and ϑ ∗ V ∆( z ) = β ∆ , ( z ), for some α, β ∈ C . By using theproperty of the adjoint map and (10), we have α k ∆ , k = h α ∆ , , ∆ , i = h ϑ ∗ ∆ , ∆ , i = h ∆ , ϑ ∆ , i = h ∆ ,
16 ∆ + 1283 V ∆ i = 16 k ∆ k + 1283 h ∆ , V ∆ i . (11)Similarly, β k ∆ , k = 1283 k V ∆ k + 16 h ∆ , V ∆ i . (12)From [4, Eq. 49], we know that h ∆ , V ∆ i = − k ∆ k . Using it in expression (11), we get α = 0, which gives ϑ ∗ ∆( z ) = 0 . (13)Now applying Theorem 4.1, we have( m − ) m τ ( m ) + 20 L ∆ ,m (11) = 0 τ ( m ) = − m ( m − ) L ∆ ,m (11) . (14)From [13, Proposition 46], we get k V ∆ k = 2 − k ∆ k and (12) gives β = 52 k ∆ k k ∆ , k . (15)Using Theorem 4.1 in the expression ∆ , ( z ) = β ϑ ∗ V ∆( z ), we get τ , ( m ) = 15 m βπ (cid:20) ( m − ) m τ (cid:16) m (cid:17) + 20 L V ∆ ,m (11) (cid:21) , (16)where τ ( n ) = 0 if n is not an integer.Therefore, for odd m , we have τ , ( m ) = 3840 m π k ∆ , k k ∆ k L V ∆ ,m (11) , where L V ∆ ,m (11) = X n > n :odd τ ( m + n ) σ ( n )( m + n ) . A. KUMAR
Remark . From (14), we see that for any m > n > τ ( m ) and τ ( m + n ) are of opposite sign. In other words, it follows that Ramanujan tau function τ ( m )and L ∆ ,m (11) both exhibit infinitely many sign changes. We can also find the values of L ∆ ,m (11) for each m >
1, in particular L ∆ , (11) = − . Moreover, Lehmer’s conjectureis equivalent to non-vanishing of L ∆ ,m (11). From (14), we also observe that for each m > , L ∆ ,m (11) ∈ Q , because the coefficient field of ∆ is Q .In general for any f ∈ S k +2 (Γ) and m >
1, using similar method, we can write L f,m ( k + 1)as a linear combination of m th Fourier coefficients of f and elements from a fixed basis of S k (Γ). Then analogous observations can be made as in Remark 5.1.6. Proof of the main Theorem
We need the following Lemma to prove the main theorem.
Lemma 6.1.
Using the same notation as in Theorem 4.1, the following sum of integrals X γ ∈ Γ ∞ \ Γ Z Γ \H | f ( z ) E ∗ ( z ) e πimz | k γ y k +2 | dxdyy converges.Proof. Since f is a cusp form of weight k , the function f E ∗ is a nearly holomorphic modularform of weight k + 2 and is rapidly decreasing at every cusp. Therefore, for some positiveconstant M, we have | y k +1 f ( z ) E ∗ ( z ) | M, ∀ z ∈ H . Changing the variable z to γ − z and using the standard Rankin unfolding argument, thesum in Lemma 6.1 equals to Z Γ ∞ \H | f ( z ) E ∗ ( z ) e πimz y k +2 | dxdyy = Z Γ ∞ \H | y k +1 f ( z ) E ∗ ( z ) | e − πmy y k +1 dxdyy M Z ∞ Z e − πmy y k − dxdy = M Γ( k )(2 πm ) k . (cid:3) Proof of Theorem 4.1.
Since ϑ ∗ k f = P m > c ( m ) q m , by Lemma 2.1, we get c ( m ) = (4 πm ) k − Γ( k − h ϑ ∗ k f, P k,m i = (4 πm ) k − Γ( k − h f, ϑ k P k,m i . HE ADJOINT MAP OF THE SERRE DERIVATIVE AND SPECIAL VALUES OF SHIFTED DIRICHLET SERIES9
By considering the above inner product in the space of nearly holomorphic modular formsand using Lemma 2.6, we get h f, ϑ k P k,m i = h f, R k P k,m − k E ∗ P k,m i = h f, R k P k,m i − k h f, E ∗ P k,m i = − k h f, E ∗ P k,m i . Hence, c ( m ) = − k
12 (4 πm ) k − Γ( k − h f, E ∗ P k,m i . (17)Now consider, h f, E ∗ P k,m i = 1 µ Γ Z Γ \H f ( z ) E ∗ ( z ) P k,m y k +2 dxdyy = 1 µ Γ Z Γ \H f ( z ) E ∗ ( z ) X γ ∈ Γ ∞ \ Γ e πimz | k γ y k +2 dxdyy . By Lemma 6.1, we can interchange summation and integration in the above expression.Using Rankin’s unfolding argument, the integral in the above expression can be written as Z Γ ∞ \H f ( z ) E ∗ ( z ) e πimz y k dxdy = Z ∞ Z X s > a ( s ) e πis ( x + iy ) − πy − X t > σ ( t ) e πit ( x + iy ) ! e πim ( x + iy ) y k dxdy = X s > a ( s ) Z ∞ Z (cid:18) − πy (cid:19) e − πy ( s + m ) y k e πix ( s − m ) dxdy − X t > σ ( t ) X s > a ( s ) Z ∞ Z e − πy ( t + m − s ) y k e πix ( s − t − m ) dxdy (18)= a ( m ) Z ∞ (cid:18) − πy (cid:19) e − πmy y k dy − X t > a ( t + m ) σ ( t ) Z ∞ e − πy ( t + m ) y k dy = Γ( k ) π (4 πm ) k (cid:18) k m − (cid:19) a ( m ) −
24 Γ( k + 1)(4 π ) k +1 X t > a ( t + m ) σ ( t )( t + m ) k +1 = Γ( k )(4 π ) k +1 (cid:20) ( k − m ) m k +1 a ( m ) − kL f,m ( k + 1) (cid:21) . By Proposition 2.2, interchanging the sum and integral in (18) is justified. Hence h f, E ∗ P k,m i = 1 µ Γ Γ( k )(4 π ) k +1 (cid:20) ( k − m ) m k +1 a ( m ) − kL f,m ( k + 1) (cid:21) . This proves the theorem.
Remark . It is worth pointing out that Lemma 2.6 holds good for forms of half-integralweight. So in view of Remark 3.2 and using the same technique as in the proof of Theorem 4.1, one can explicitly find the map ϑ ∗ k/ : S k +2 (Γ) −→ S k (Γ) , where k is an odd positive integer and Γ = Γ ( N ) , N ∈ N . It gives a construction of cuspforms of half-integral weight whose coefficients involve special values of shifted Dirichletseries of Rankin type. Acknowledgements.
I would like to thank Dr. B. Sahu for raising this question andfor some useful discussions. I am grateful to my supervisor Prof. B. Ramakrishnan for hisconstant support, comments and suggestions about the paper. The work has been supportedby the SPM research grant of the Council of Scientific and Industrial Research (CSIR), India.
References [1] K. Bringmann, M. H. Mertens and K. Ono, p -adic properties of modular shifted convolution Dirichletseries , Proc. Amer. Math. Soc. (2016) 1439–1451.[2] J. H. Bruinier, Borcherds products of O (2 , l ) and Chern classes of Heegner divisors, Lect. Notes Math. , Springer , Berlin, 2002.[3] J. H. Bruinier, G. van der Geer, G. Harder and D. Zagier, The 1-2-3 of Modular Forms,
Springer , Berlin,2008.[4] F. Chiera and K. Vankov,
On special values of spinor L-functions of Siegel cusp eigenforms of genus 3 ,arXiv:0805.2114v1, 2008.[5] Y. Choie, H. Kim and M. Knopp,
Construction of Jacobi forms,
Math. Z. (1995) 71–76.[6] P. Deligne,
La conjecture de Weil. I. (French), Inst. Hautes tudes Sci. Publ. Math. (1974) 273–307.[7] S. D. Herrero, The adjoint of some linear maps constructed with the Rankin-Cohen brackets , RamanujanJ. (3) (2015) 529-536.[8] J. Hoffstein and T. A. Hulse, Multiple Dirichlet series and shifted convolution , Journal of NumberTheory (2016) 417–533.[9] A. K. Jha and A. Kumar,
Construction of cusp forms using Rankin-Cohen brackets , arXiv:1607.03511v1,2016.[10] A. K. Jha and B. Sahu,
Rankin-Cohen brackets on Jacobi forms and the adjoint of some linear maps ,Ramanujan J. (3) (2016) 533-544.[11] A. K. Jha and B. Sahu, Rankin-Cohen brackets on Siegel modular forms and special values of certainDirichlet series , Ramanujan J. DOI: 10.1007/s11139-016-9783-3, 2016.[12] M. Kaneko and D. Zagier,
A generalized Jacobi theta function and quasimodular forms,
The modulispace of curves (Texel Island, 1994), Progr. Math., 129, Birkhuser Boston, Boston, MA (1995) 165–172.[13] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, 2nd edn., Graduate Texts in Mathe-matics , Springer-Verlag , New York, 1993.[14] W. Kohnen,
Cusp forms and special value of certain Dirichlet series , Math. Z. (1991) 657–660.[15] M. H. Lee,
Siegel cusp forms and special values of Dirichlet series of Rankin type , Complex Var. TheoryAppl. (2) (1996) 97–103.[16] M. H. Mertens and K. Ono, Special values of shifted convolution Dirichlet series , Mathematika (1)(2016) 47–66.[17] K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q -Series, CBMSRegional Conference Series in Mathematics, , American Mathematical Society , Providence, RI, 2004.[18] H. Sakata,
Construction of Jacobi cusp forms , Proc. Japan. Acad. Ser. A Math. Sci. (7) (1998)117–119. HE ADJOINT MAP OF THE SERRE DERIVATIVE AND SPECIAL VALUES OF SHIFTED DIRICHLET SERIES11 [19] A. Selberg,
On the estimation of Fourier coefficients of modular forms , Proc. Sympos. Pure Math.,VIII, Amer. Math. Soc., Providence, RI, (1965) 1–15.[20] G. Shimura, Modular forms: Basics and Beyond, Springer Monographs in Mathematics,
Springer , NewYork, 2012.[21] X. Wang,
Hilbert modular forms and special values of some Dirichlet series , Acta. Math. Sin. (3)(1995) 336–343. Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211019,India.
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