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.     

Poles of Eisenstein series on quaternion groups

We show that the normalized Siegel Eisenstein series of quaternion groups have at most simple poles at certain integers and half integers. These Eisenstein series play an important role of Rankin-Selberg integral representations of Langlands L-functions for quaternion groups.     

Computations of Eisenstein series on Fuchsian groups

We present numerical investigations of the value distribution and distribution of Fourier coefficients of the Eisenstein series on arithmetic and non-arithmetic Fuchsian groups. Our numerics indicate a Gaussian limit value distribution for a real-valued rotation of as , and also, on non-arithmetic groups, a complex Gaussian limit distribution for when near and , at least if we allow at some rate. Furthermore, on non-arithmetic groups and for fixed with near , our numerics indicate a Gaussian limit distribution for the appropriately normalized Fourier coefficients.

     

Theta functions and symplectic groups

A. Weil introduced in 1964 theta functions on locally compact commutative groups (cf.[5]III). It is shown here that the use of certain function spaces on such a groupG, and the consideration of theta functions on the productG×G, gives some insight into the structures involved, also in connection with Poisson's formula.     

Multiple Eisenstein series and multiple cotangent functions

We construct the multiple Eisenstein series and we show a relation to the multiple cotangent function. We calculate a limit value of the multiple Eisenstein series.     

Corrigendum Eisenstein series for Siegel modular groups

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

Constant terms of spherical Eisenstein series on quasi-split unitary groups

In this note, we prove a formula which expresses the constant term of the spherical Eisenstein series on a quasi-split unitary group as a linear combination of spherical Eisenstein series on smaller unitary groups.     

On Eisenstein polynomials and zeta polynomials

Eisenstein polynomials, which were defined by Oura, are analogues of the concept of an Eisenstein series. Oura conjectured that there exist some analogous properties between Eisenstein series and Eisenstein polynomials. In this paper, we provide new analogous properties of Eisenstein polynomials and zeta polynomials. These properties are finite analogies of certain properties of Eisenstein series.     

Limit values of Eisenstein series and multiple cotangent functions

We show a limit formula for Eisenstein series by using the theory of a multiple cotangent function. The value is expressed simply via the Bernoulli number.     

Ramanujan series for Epstein zeta functions

In the spirit of Ramanujan, we derive exponentially fast convergent series for Epstein zeta functions \(E^{\varGamma _0(N)}(z,s)\) on the Hecke congruence groups \( \varGamma _0(N),N\in \mathbb {Z}_{>0}\), where z is an arbitrary point in the upper half-plane \( \mathfrak {H}\) and \(s\in \mathbb {Z}_{>1}\). These Ramanujan series can be reformulated as integrations of modular forms, in the framework of Eichler integrals. Particular cases of these Eichler integrals recover part of the recent results reported by Wan and Zucker (arXiv:1410.7081v1, 2014).     

Eisenstein series and theta functions to the septic base

We develop a theory for Eisenstein series to the septic base, which was started by S. Ramanujan in his “Lost Notebook.” We show that two types of septic Eisenstein series may be parameterized in terms of the septic theta function and the eta quotient η⁴(7τ)/η⁴(τ). This is accomplished by constructing elliptic functions which have the septic Eisenstein series as Taylor coefficients. The elliptic functions are shown to be solutions of a differential equation, and this leads to a recurrence relation for the septic Eisenstein series.     

Double zeta values, double Eisenstein series, and modular forms of level 2

We study the double shuffle relations satisfied by the double zeta values of level 2, and introduce the double Eisenstein series of level 2 which satisfy the double shuffle relations. We connect the double Eisenstein series to modular forms of level 2.     

Telescoping, rational-valued series, and zeta functions

We give an effective procedure for determining whether or not a series telescopes when is a rational function with complex coefficients. We give new examples of series , where is a rational function with integer coefficients, that add up to a rational number. Generalizations of the Euler phi function and the Riemann zeta function are involved. We give an effective procedure for determining which numbers of the form are rational. This procedure is conditional on 3 conjectures, which are shown to be equivalent to conjectures involving the linear independence over the rationals of certain sets of real numbers. For example, one of the conjectures is shown to be equivalent to the well-known conjecture that the set is linearly independent, where is the Riemann zeta function.

Some series of the form , where is a quotient of symmetric polynomials, are shown to be telescoping, as is . Quantum versions of these examples are also given.

     

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