On the analytic continuation of Eisenstein series for Siegel's modular group of degreen

For \(M = \left( {\begin{array}{*{20}c} {A B} \\ {C D} \\ \end{array} } \right)\) ∈ Γ(n)=Sp(n?) andZ=Z+iY,Y > 0, set $$M\left\langle Z \right\rangle = (AZ + B)(CZ + D)^{ - 1} = X_M + iY_M ;M\{ Z\} = CZ + D.$$ Denote with Γ _n (n) the subgroup defined byC=0. Forr∈? and a complex variable ω form the Eisenstein series $$E(n,r,Z,\omega ) = \sum\limits_{M\varepsilon I'_n (n)\backslash \Gamma (n)} {(DetM\{ Z\} )^{ - 2r} (DetY_M )^{\omega - r} } .$$ It is proved thatE(n, r, Z, ω) can be meromorphically continued to the ω-plane and satisfies a functional equation. Forr=1, 2, [(n?1)/2], [(n+1)/2] the functionE(n, r, Z, ω) is holomorphic at ω-r. For 3≤r≤[(n?3)/2] the functionE(n, r, Z, ω) may have poles at ω=r. But the pole-order is for two smaller than known until now. This result says especially that the Eisenstein series has Hecke summation forr=1, 2, [(n?1)/2], [(n+1)/2].     

Real zeros of Eisenstein series and Rankin-Selberg L-functions

We prove that the Eisenstein series E(z, s) have no real zeroes for s ∈ (0, 1) when the value of the imaginary part of z is in the range $\tfrac{1}{5}$ < Im z < 4.94. For very large and very small values of the imaginary part of z, E(z, s) have real zeros in (½, 1), i.e. GRH does not hold for the Eisenstein series. Using these properties, we prove that the Rankin-Selberg L-function attached with the Ramanujan τ-function has no real zeros in the critical strip, except at the central point s = ½.     

The Rankin-Selberg convolution for real analytic Cohen's Eisenstein series of half-integral weight

We give a meromorphic continuation and a functional equationfor the Rankin–Selberg convolution of certain real analyticEisenstein series of half-integral weight. Our result and methodhave several applications to the Koecher–Maass seriesassociated with the real analytic Siegel–Eisenstein series.     

On differential modular forms and some analytic relations between Eisenstein series

In the present article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight 2, 4 and 6. We define Hecke operators on them, find some analytic relations between these Eisenstein series and obtain them in a natural way as coefficients of a family of elliptic curves. The fact that a complex manifold over the moduli of polarized Hodge structures in the case h ¹⁰=h ⁰¹=1 has an algebraic structure with an action of an algebraic group plays a basic role in all of the proofs.      

Sixteen Eisenstein series

S. Ramanujan gave fourteen families of series in his Second Notebook in Chap. 17, Entries 13–17. In each case he gave only the first few examples, giving us the motivation to find and prove a general formula for each family of series. The aim of this paper is to develop a powerful tool (four versatile functions f ₀,f ₁,f ₂, and f ₃) to collect all of Ramanujan’s examples together.      

Fourier coefficients of real analytic Eisenstein series at various cusps (II)

The Ramanujan Journal - The purpose of this paper is to consider the computation of the Fourier coefficients of real-analytic Eisenstein series associated to general cusps of Hecke congruence...     

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.     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

Eisenstein series on the symplectic group

Analytic continuation is proved for certain Eisenstein series on the symplectic group which are associated with nonparabolic forms. In the case of the full modular group an explicit functional equation is obtained, and the singularities of the series are completely described.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 69, pp. 103–105, 1977.The author is grateful to Prof. A. N. Andrianov for posing the problem and for his constant attention to the work.     

Eisenstein series for SL(2)

This paper gives explicit formulas for the Fourier expansion of general Eisenstein series and local Whittaker functions over SL₂. They are used to compute both the value and derivatives of these functions at critical points.     

On the Jacobi series

Summary The Jacobi series of a functionf is an expansion in a series of ascending powers of a prescribed polynomialP of degreen in which the coefficients are polynomials of lesser degree. These coefficients are usually expressed as contour integrals or are determined by their interpolatory properties. We show how they may be expressed as generalized derivatives off with respect toP. In so doing we also show how the Jacobi series may be expressed (in yet another way) as a generalized Taylor series. In addition, we obtain a number of interesting relations among the generalized derivatives.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

Corrigendum Eisenstein series for Siegel modular groups

Math. Ann. 297, 581-625 (1993) Received: 9 February 1996 / Revised version: 29 August 1996     

Eisenstein series and convolution sums

The Ramanujan Journal - We compute Fourier series expansions of weight 2 and weight 4 Eisenstein series at various cusps. Then we use results of these computations to give formulas for the...     

Transformation theory of Eisenstein series

Mathematische Annalen -