Taylor expansion of an Eisenstein series

In this paper, we give an explicit formula for the first two terms of the Taylor expansion of a classical Eisenstein series of weight for . Both the first term and the second term have interesting arithmetic interpretations. We apply the result to compute the central derivative of some Hecke -functions.

     

Eisenstein series associated with Γ<Subscript>0</Subscript>(2)

In this paper, we define the normalized Eisenstein series ℘, e, and associated with Γ₀(2), and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas, we define a differential equation depending on the weights of modular forms on Γ₀(2) and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from a formula involving the triangular numbers.      

Representations of Integers as Sums of 32 Squares

In this paper, we derive a new explicit formula for r ₃₂(n), where r _k(n) is the number of representations of n as a sum of k squares. For a fixed integer k, our method can be used to derive explicit formulas for r _8k (n). We conclude the paper with various conjectures that lead to explicit formulas for r _2k (n), for any fixed positive integer k > 4.     

Linear relations of theta series of genera of quadratic forms

In this paper we study linear relations among theta series of genera of positive definite n-ary quadratic forms with given level D,2D,4D and 8D for square free D. We obtain a basis for the space generated by genus theta series. This forms a basis of Eisenstein space.     

On -adic Hermitian Eisenstein series

In this paper we generalize the notion of -adic modular form to the Hermitian modular case and prove a formula that shows a coincidence between certain -adic Hermitian Eisenstein series and the genus theta series associated with Hermitian matrix with determinant .

     

A generalization of the Lipschitz summation formula and some applications

The Lipschitz formula is extended to a two-variable form. While the theorem itself is of independent interest, we justify its existence further by indicating several applications in the theory of modular forms.

     

On singular modular forms for principal congruence subgroups

We show that spaces of vector–valued singular modular forms for principal congruence subgroups of the symplectic group Sp(n,ℤ) of integral weight are generated by suitable finite dimensional families of Siegel theta series. This is obtained as an application of some results concerning the action of trace operators on non–homogeneous theta series.     

Generalized Dedekind symbols associated with the Eisenstein series

We have shown recently that the space of modular forms, the space of generalized Dedekind sums, and the space of period polynomials are all isomorphic. In this paper, we will prove, under these isomorphisms, that the Eisenstein series correspond to the Apostol generalized Dedekind sums, and that the period polynomials are expressed in terms of Bernoulli numbers. This gives us a new more natural proof of the reciprocity law for the Apostol generalized Dedekind sums. Our proof yields as a by-product new polylogarithm identities.

     

On a unit group generated by special values of Siegel modular functions

There has been important progress in constructing units and -units associated to curves of genus 2 or 3. These approaches are based mainly on the consideration of properties of Jacobian varieties associated to hyperelliptic curves of genus 2 or 3. In this paper, we construct a unit group of the ray class field of modulo 6 with full rank by special values of Siegel modular functions and circular units. We note that . Our construction of units is number theoretic, and closely based on Shimura's work describing explicitly the Galois actions on the special values of theta functions.

     

Ramanujan's Eisenstein series and new hypergeometric-like series for

Using hypergeometric identities and certain representations for Eisenstein series, we uniformly derive several new series representations for 1/π².     

Spectral analysis and the Riemann hypothesis

The explicit formulas of Riemann and Guinand-Weil relate the set of prime numbers with the set of nontrivial zeros of the zeta function of Riemann. We recall Alain Connes’ spectral interpretation of the critical zeros of the Riemann zeta function as eigenvalues of the absorption spectrum of an unbounded operator in a suitable Hilbert space. We then give a spectral interpretation of the zeros of the Dedekind zeta function of an algebraic number field K of degree n in an automorphic setting.

If K is a complex quadratic field, the torical forms are the functions defined on the modular surface X, such that the sum of this function over the “Gauss set” of K is zero, and Eisenstein series provide such torical forms.

In the case of a general number field, one can associate to K a maximal torus T of the general linear group G. The torical forms are the functions defined on the modular variety X associated to G, such that the integral over the subvariety induced by T is zero. Alternately, the torical forms are the functions which are orthogonal to orbital series on X.

We show here that the Riemann hypothesis is equivalent to certain conditions bearing on spaces of torical forms, constructed from Eisenstein series, the torical wave packets. Furthermore, we define a Hilbert space and a self-adjoint operator on this space, whose spectrum equals the set of critical zeros of the Dedekind zeta function of K.     

Equilibrium point of Green's function for the annulus and Eisenstein series

We study the motion of the equilibrium point of Green's function and give an explicit parametrization of the unique zero of the Bergman kernel of the annulus. This problem is reduced to solving the equation , where is the usual Eisenstein series.

     

A note on the non-holomorphic Eisenstein series

We give a sufficient condition of bounded growth for the non-holomorphic Eisenstein series on SL ₂(ℤ). The C ^∞-automorphic forms of bounded growth are introduced by Sturm (Duke Math. J. 48(2), 327–350, 1981) in the study of automorphic L-functions. We also give a Laplace-Mellin transform of the Fourier coefficients of the Eisenstein series. The transformation constructs a projection of the Eisenstein series to the space of holomorphic cusp forms.      

On zeros of Eisenstein series for genus zero Fuchsian groups

Let SL be a genus zero Fuchsian group of the first kind with as a cusp, and let be the holomorphic Eisenstein series of weight on that is nonvanishing at and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on and on a choice of a fundamental domain , we prove that all but possibly of the nontrivial zeros of lie on a certain subset of . Here is a constant that does not depend on the weight, is the upper half-plane, and is the canonical hauptmodul for

     

On Eisenstein series and

In this paper, we derive some new identities satisfied by the series using Ramanujan's identities for , and . Our work is motivated by an attempt to develop a theory of elliptic functions to the septic base.

     

Eisenstein Series in Ramanujan's Lost Notebook

In his lost notebook, Ramanujan stated without proofs several beautifulidentities for the three classsical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.