Fourier Coefficients Of Cusp Forms Research Articles

Fourier coefficients of cusp forms and automorphic f-functions

In the beginning of the paper, we review the summation formulas for the Fourier coefficients of holomorphic cusp forms for γ, γ=SL(2,ℤ), associated with L-functions of three and four Hecke eigenforms. Continuing the known works on the L-functions Lf,ϕ,ψ(s) of three Hecke eigenforms, we prove their new properties in the special case of Lf,ϕ,ψ(s). These results are applied to proving an analogue of the Siegel theorem for the L-function Lf(s) of the Hecke eigenform f(z) for γ (with respect to weight) and to deriving a new summation formula. Let f(z) be a Hecke eigenform for γ of even weight 2k with Fourier expansion\(f(z) = \Sigma _{n = 1}^\infty a(n)e^{2\pi inz} \). We study a weight-uniform analogue of the Hardy problem on the behavior of the sum ⌆p≤xa(p) log p and prove new estimates from above for the sum ⌆n≤xa(F(n))2, where F(x) is a polynomial with integral coefficients of special form (in particular, F(x) is an Abelian polynomial). Finally, we obtain the lower estimate $$L_4 (1) + |L'_4 (1)| \gg \frac{1}{{(\log k)^c }},$$ where L4(s) is the fourth symmetric power of the L-function Lf(s) and c is a constant. Bibliography: 43 titles.

Sobolev norms of automorphic functionals and Fourier coefficients of cusp forms

We propose a new approach to the study of eigenfunctions of the Laplace-Beltrami operator on a Riemann surface of curvature 1. It is based on Frobenius reciprocity from the theory of automorphic functions. We determine Sobolev class of arising automorphic functionals and discuss some applications.

On twisting operators and newforms of half-integral weight II – Complete theory of newforms for Kohnen space

Abstract.The author continues his previous work with the purpose of establishing a theory of newforms in the case of half-integral weight. In this paper, the author formulates and proves a complete theory of newforms for Kohnen space. Kohnen space is an important canonical subspace in the space of cusp forms of half-integral weight k + 1/2 (k > 0). Every Hecke common eigenform in Kohnen space corresponds to a primitive form of integral weight 2k and of odd level via Shimura Correspondence.These newforms for Kohnen space satisfy the Strong multiplicity One theorem. Moreover, we explicitly determine the corresponding primitive form (of weight 2k) to each newform for Kohnen space. The space of oldforms is also explicitly described.In order to find all newforms for Kohnen space, the author needs a certain non-vanishing property of Fourier coefficients of cusp forms. Such property proves by using representation theory of finite linear groups. The method of proof of newform theory is mainly based on trace formulae and trace relations.

On the third and fourth power moments of Fourier coefficients of cusp forms

The asymptotic formulae for the third and fourth power moments of Fourier coefficients of cusp forms are proved in this paper.

The additive divisor problem and its analogs for Fourier coefficients of cusp forms. II

The additive divisor problem and its analogs for Fourier coefficients of cusp forms. II

On the distribution and moments of the fourier coefficients of cusp forms

Two theorems about the distribution and moments of the Fourier coefficients of cusp forms are proved in this paper.

The additive divisor problem and its analogs for fourier coefficients of cusp forms. I

The additive divisor problem and its analogs for fourier coefficients of cusp forms. I

The additive divisor problem and its analogs for fourier coefficients of cusp forms. I

The additive divisor problem and its analogs for fourier coefficients of cusp forms. I

Applications of the petersson formula for a bilinear form in fourier coefficients of cusp forms

Let Sk(Γ0(N),χ) be the space of holomorphic Γ0(N) forms of integral weight k and character χ. Let fj(z), 1≤j≤v new2k (p), be the set of normalized newforms of S2k(Γ0(p),1), where p is a prime, and let\(L_j (s) = L_{f_i } (s)\) be the L-function of fj(z). It is proved that $$\sum\limits_{1 \leqslant j \leqslant v_{2k}^{new} (p)} {L_j^2 (\tfrac{1}{2}) \ll p\log ^4 p \cdot \log \log p} , p \to \infty$$ where 2k≥4. Errors in an earlier paper (RŽMat, 1989, 4A65) are corrected. Bibliography: 11 titles.

Third power moments of the error terms corresponding to certain arithmetic functions

For positive integersn, letd(l 1,M 1;l 2,M 2;n) denote the number of factorizationsn=n 1 n 2 where each of the factorsn∈ℕ belongs to a prescribed congruence classl i moduloM i (i=1,2). In this article an asymptotic result is derived for the third power moment of the error term in the formula for the Dirichlet summmatory function ofd(l 1,M 1;l 2,M 2;n). This extends a recent result of [17] for the classic “unrestricted” case ofd(n)=d(1,1;1,1; n). Moreover, the technique developed is applied to the analogous problem related to Fourier coefficients of cusp forms.

Kloosterman Sums and a Mean Value for Dirichlet Polynomials

Let a 1, a 2, a 3,... be complex numbers. Then, for ϵ > 0, M ≥ 1 and T ≥ 1, [formula] where ζ( s) is Riemann′s zeta-function and C ϵ depends only on ϵ. The proof is based on a technical refinement within the circle of ideas to be found in Deshouillers and Iwaniec′s paper: Kloosterman sums and Fourier coefficients of cusp forms ( Invent. Math. 70 (1982), 219-288).

On Zagier’s cusp form and the Ramanujan τ function

Zagier constructed a cusp form for each weightk of the full modular group. We use this construction to estimate Fourier coefficients of cusp forms. In particular, we get a non-trivial estimate, by elementary methods and indicate a relationship with the Lindelof hypothesis for classical Dirichlet L-functions.

On a property of Fourier coefficients of cusp forms of half-integral weight

On a property of Fourier coefficients of cusp forms of half-integral weight

Ω-results for sums of Fourier coefficients of cusp forms

Ω-results for sums of Fourier coefficients of cusp forms

On Whittaker models and the vanishing of Fourier coefficients of cusp forms

The purpose of this paper is to construct examples of automorphic cuspidal representations which possess a ψ-Whittaker model even though their ψ-Fourier coefficients vanish identically. This phenomenon was known to be impossible for the groupGL(n), but in general remained an open problem. Our examples concern the metaplectic group and rely heavily upon J L Waldspurger’s earlier analysis of cusp forms on this group.

Fourier coefficients of cusp forms

The object of this survey article is to trace the influence on the theory of modular forms of the ideas contained in L. J. Mordell's important paper ‘On Mr Ramanujan's empirical expansions of modular functions’, which appeared in October 1917 in this Society'sProceedings[32]. The equally important paper [42] by S. Ramanujan, ‘On certain arithmetical functions’, referred to in Mordell's title, was published in May 1916 in the same Society's older journal, theTransactions, which was regrettably suppressed in 1928, 107 years after its foundation. Ramanujan's paper was concerned not only with multiplicative properties of Fourier coefficients of modular forms, but also with their order of magnitude. Since subsequent papers on the latter subject have also appeared in theProceedings, it seems appropriate to include further developments in this field of study in the present survey.

On various means involving the Fourier coefficients of cusp forms

On various means involving the Fourier coefficients of cusp forms

Kloosterman sums and Fourier coefficients of cusp forms

Kloosterman sums and Fourier coefficients of cusp forms

Convolutions of Dirichlet series with the Fourier coefficients of cusp forms of weight 0

By Rankin's method one constructs a meromorphic continuation and a functional equation for the convolution of two Dirichlet series corresponding to cusp forms of weight O. One investigates the summator function of the coefficients of this convolution.

Fourier coefficients of cusp forms

Let ψ1,ψ2,ψ3,... be an orthonormal basis of the space of cusp forms of weight zero for the full modular group. Let be the Fourier series expansion. The following theorem is proved: Let σ∈(1/4, 1/2); letf be a holomorphic function on the strip |Res|≦σ, satisfyingf(−s)=f(s) and $$f(s) = \mathcal{O}(|\tfrac{1}{4} - s^2 |^{ - 2} |cos \pi s|^{ - 1} )$$ on this strip; letm andn be non-zero integers, then $$\sum\limits_{j = 1}^\infty {f(s_j )\bar \gamma _{jm} \gamma _{jn} } $$ converges and is equal to $$\begin{gathered} - (2\pi i)^{ - 1} \int\limits_{\operatorname{Re} s = 0} {f(s)c_{00} ( - s)c_{0|m|} (s)c_{0|n|} (s)ds} \hfill \\ + (2\pi i)^{ - 1} (4\pi |m|)^{ - 1} \int\limits_{\operatorname{Re} s = 0} {f(s)c_{mn} (s)2sds} \hfill \\ - \delta _{mn} (2\pi i)^{ - 1} (4\pi |m|)^{ - 1} \int\limits_{\operatorname{Re} s = 0} {f(s)\sin \pi s2sds.} \hfill \\ \end{gathered} $$