Transformation formulae and asymptotic expansions for double holomorphic Eisenstein series of two complex variables

The main object of study in this paper is the double holomorphic Eisenstein series \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) having two complex variables \(\mathbf{s}=(s_1,s_2)\) and two parameters \(\mathbf{z}= (z_1,z_2)\) which satisfies either \(\mathbf{z}\in (\mathfrak {H}^+)^2\) or \(\mathbf{z}\in (\mathfrak {H}^-)^2\), where \(\mathfrak {H}^{\pm }\) denotes the complex upper and lower half-planes, respectively. For \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\), its transformation properties and asymptotic aspects are studied when the distance \(|z_2-z_1|\) becomes both small and large under certain natural settings on the movement of \(\mathbf{z}\in (\mathfrak {H}^{\pm })^2\). Prior to the proofs our main results, a new parameter \(\eta \), which plays a pivotal role in describing our results, is introduced in connection with the difference \(z_2-z_1\). We then establish complete asymptotic expansions for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) when \(\mathbf{z}\) moves within the poly-sector either \((\mathfrak {H}^+)^2\) or \((\mathfrak {H}^-)^2\), so as to \(\eta \rightarrow 0\) through \(|\arg \eta |<\pi /2\) in the ascending order of \(\eta \) (Theorem 1). This further leads us to show that counterpart expansions exist for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) in the descending order of \(\eta \) as \(\eta \rightarrow \infty \) through \(|\arg \eta |<\pi /2\) (Theorem 2). Our second main formula in Theorem 2 yields a functional equation for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) (Corollaries 2.1, 2.2), and also reduces naturally to various expressions of \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) in closed forms for integer lattice point \(\mathbf{s}\in \mathbb {Z}^2\) (Corollaries 2.3–2.17). Most of these results reveal that the particular values of \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) at \(\mathbf{s}\in \mathbb {Z}^2\) are closely linked to Weierstraß’ elliptic function, the classical Eisenstein series reformulated by Ramanujan, and the Jordan–Kronecker type functions, each associated with the bases \(2\pi (1, z_j)\), \(j=1,2\). The latter two functions were extensively utilized by Ramanujan in the course of developing his theories of Eisenstein series, elliptic functions, and theta functions. As for the methods used, crucial roles in the proofs are played by the Mellin–Barnes type integrals, manipulated with several properties of hypergeometric functions; the transference from Theorem 1 to Theorem 2 is, for instance, achieved by a connection formula for Kummer’s confluent hypergeometric 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.     

An asymptotic series for an integral

The Ramanujan Journal - Let $$f(z)=\sum _{n=0}^{\infty }a_{n}{\mathbf {e}}(nz),g(z)=\sum _{n=0}^{\infty }b_{n}{\mathbf {e}}(nz)\ ({\mathbf {e}}(z)=e^{2\pi \sqrt{-1}z})$$ be holomorphic modular...     

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.      

On an asymptotic series and corresponding continued fraction for a gamma function ratio

Recurrence relations for the coefficients in the asymptotic expansion of a gamma function ratio are derived and a property of these coefficients is proved. The Stieltjes fraction for the series is given and a characteristic of the partial numerators is explained. A connection between the continued fraction and the error of a particular least squares approximation problem is discussed.     

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...     

A theta function identity of degree eight and Eisenstein series identities

In this paper we prove a theta function identity of degree eight using the theory of elliptic theta functions and the method of asymptotic analysis. This identity allows us to derive some curious Eisenstein series identities. We prove a new addition formula for theta functions which allows us to give an extension of the Hirschhorn septuple product identity.     

Quintic and septic Eisenstein series

Using results that were well-known to Ramanujan, we give proofs of some results for Eisenstein series in the lost notebook. Our proofs have the additional advantage that it is not necessary to know the results in advance; that is, the proofs are derivations as opposed to verifications.      

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).     

Arithmetic of quaternions and Eisenstein series

In the paper one computes the Fourier coefficients of the Eisenstein series of the orthogonal group of signature (1, 4). The formulas show that the restriction of the Eisenstein series to the “imaginary” axis is a Dirichlet series, whose coefficients are the products of the L-series by the number of the representations of the given number as a sum of three squares. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 82–90, 1987.     

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.     

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.