On the Fourier coefficients of Siegel–Eisenstein series of degree 2 for odd level

The Ramanujan Journal - We calculate the Fourier coefficients of the Siegel–Eisenstein series of degree 2, level $$p^n$$ with trivial character. The main result is to compute the Siegel...     

On Fourier–Jacobi expansions of real analytic Eisenstein series of degree 2

We discuss the Fourier–Jacobi expansion of certain vector valued Eisenstein series of degree $2$ , which is also real analytic. We show that its coefficients of index $\pm 1$ can be described by using a generating series of real analytic Jacobi forms. We also describe all the coefficients of general indices in suitable manners. Our method can be applied to study another Fourier series of Saito-Kurokawa type that is associated with a cusp form of one variable and half-integral weight. Then, following the arguments in the holomorphic case, we find that the Fourier series indeed defines a real analytic Siegel modular form of degree 2.     

Projections of modular forms on Eisenstein series and its application to Siegel’s formula

The Ramanujan Journal - Let $$k ge 2$$ and N be positive integers and let $$chi $$ be a Dirichlet character modulo N. Let f(z) be a modular form in $$M_k(Gamma _0(N),chi )$$ . Then we have a...     

Nonvanishing of Siegel Poincaré series

We prove that, under suitable conditions, certain Siegel Poincaré series of exponential type of even integer weight and degree 2 do not vanish identically. We also find estimates for twisted Kloosterman sums and Salié sums in all generality.     

Nonvanishing of Koecher–Maass series attached to Siegel cusp forms

We prove a nonvanishing result for Koecher–Maass series attached to Siegel cusp forms of weight k and degree n   in certain strips on the complex plane. When n=2$n=2$, we prove such a result for forms orthogonal to the space of the Saito–Kurokawa lifts ‘up to finitely many exceptions’, in bounded regions.     

The McKay–Thompson series of Mathieu Moonshine modulo two

In this note, we describe the parity of the coefficients of the McKay–Thompson series of Mathieu moonshine. As an application, we prove a conjecture of Cheng, Duncan, and Harvey stated in connection with umbral moonshine for the case of Mathieu moonshine.     

Differential equations satisfied by Eisenstein series of level 2

Ramanujan’s differential equations for the classical Eisenstein series are of great importance to many areas in number theory and special functions. H.H. Chan recently demonstrated that these differential equations can be derived from the triple product identity and the quintuple product identity in an elementary manner. In this article, we extend this method in a uniform manner to derive corresponding differential equations for the Eisenstein series of level 2. Several applications of these differential equations are also given.     

Eisenstein series and Ramanujan-type series for 1/π

Using certain representations for Eisenstein series, we uniformly derive several Ramanujan-type series for 1/π.     

Rankin–Cohen brackets on Siegel modular forms and special values of certain Dirichlet series

Given a fixed Siegel cusp form of genus two, we consider a family of linear maps between the spaces of Siegel cusp forms of genus two by using the Rankin–Cohen brackets and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Siegel cusp forms of genus two constructed using this method involve special values of certain Dirichlet series of Rankin type associated to Siegel cusp forms. This is a generalisation of the work due to Kohnen (Math Z 207:657–660, 1991) and Herrero (Ramanujan J 36:529–536, 2015) in the case of elliptic modular forms to the case of Siegel cusp forms which is also considered earlier by Lee (Complex Var Theory Appl 31:97–103, 1996) for a special case.     

A study on twisted Koecher–Maass series of Siegel cusp forms via an integral kernel

The Ramanujan Journal - The authors wish to draw the attention to a mistake which appears in the proof of Proposition 3 of the above quoted paper [4].     

Products of two 1ϕ1 series and implications

An expansion formula into bivariate formal power series is established that implies, as consequences, three remarkable identities of partial theta functions due to Warnaar (2003), Schilling–Warnaar (2002) and Andrews–Warnaar (2007).     

Fibers of the Baum-Bott map for foliations of degree two on ?2

The Baum-Bott map associates to a foliation the Baum-Bott indexes of their singularities. In this paper we study the fibers of the Baum-Bott map in the space of foliations of degree two on the projective plane ℙ². In the main result we prove that its generic fiber contains exactly 240 orbits of the natural action of Aut(ℙ²) onthespace of foliations.     

Triangular Ces??ro summability of two dimensional Fourier series

It is proved that the maximal operator of the triangular Ces??ro means of a two-dimensional Fourier series is bounded from the periodic Hardy space $H_{p}(\mathbb{T}^{2})$ to $L_{p}(\mathbb{T}^{2})$ for all 2/(2+??)<p?Q?? and, consequently, is of weak type (1,1). As a consequence we obtain that the triangular Ces??ro means of a function $f \in L_{1}(\mathbb{T}^{2})$ converge a.e. to?f.     

Universality properties of Walsh–Fourier series

It is shown that Walsh–Fourier series of \(W\) -continuous functions can have maximal sets of limit functions on small subsets of the unit interval.     

A Generalization of the Siegel–Walfisz Theorem and its Application

We investigate the distribution of integers with a fixed number of prime factors in arithmetic progressions, and obtain a generalization of the Siegel–Walfisz theorem under the extended Riemann hypothesis. As an application, we consider a problem of P. Erd?s, A. M. Odlyzko and A. Sárközy about the representation of residue classes modulo q by products of two integers with a fixed number of prime factors. We show some conditional results.