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A converse theorem for double Dirichlet series and Shintani zeta functions. (English) Zbl 1311.11043

Let \(N\) and \(D\) be positive integers with \((D,4N)=1\). Let \(v\) be the weight \(1/2\) multiplier system of the congruence subgroup \(\Gamma_0(4N)\) defined as \[ v(\gamma)=\left(\frac{c}{d}\right)\varepsilon_d^{-1}\quad \text{for\;} \gamma=\begin{pmatrix} a & b\\ c & d \end{pmatrix}, \] with \(\left(\frac{c}{d}\right)\) the usual Kronecker symbol and \[ \varepsilon_d=\left\{\begin{matrix} 1, & d\equiv 1\pmod 4,\\ i, & d\equiv 3\pmod 4. \end{matrix}\right. \] Denote by \(m^{\ast}\) the number of inequivalent cusps of \(\Gamma_0(4N)\) which are singular with respect to \(v\). For complex numbers \(a_{n,l}^j\) (with \(n,l\in\mathbb{Z}, l\geqslant 1, j=1,2,\ldots,m^{\ast}\)) and a Dirichlet character \(\chi\pmod D\), the double Dirichlet series \[ L_j^{\pm}(s,w;\chi)=\sum\limits_{\pm n>0}\sum\limits_{l=1}^\infty\frac{a_{n,l}^j\tau_n(\chi)}{l^w|n|^s} \] with \(s,w\in\mathbb{C}\) are considered in this paper, where \(\tau_n(\chi)\) is the Gauss sum. A converse theorem for the family of these double Dirichlet series is proved, which states that if the family satisfies certain “nice properties”, then the family consists of linear combinations of Mellin transforms of metaplectic Eisenstein series. For the case of \(\Gamma_0(4)\), the scalar version of the converse theorem is obtained, which is much more simpler. Using this simpler converse theorem, the author show that Shintani’s zeta functions are Mellin transforms of the metaplectic Eisenstein series for \(\Gamma_0(4)\). Here Shintani’s zeta functions are defined by \[ \xi_j(s_1,s_2)=2^{-1}\sum\limits_{n,m=1}^\infty A(4m,(-1)^{j-1}n)m^{-s_1}n^{-s_2} \] and \[ \xi_j^{\ast}(s_1,s_2)=\sum\limits_{n,m=1}^\infty A(m,(-1)^{j-1}n)m^{-s_1}(4n)^{-s_2}, \] with \(j=1,2\), \(\operatorname{Re}(s_1),\operatorname{Re}(s_2)>1\) and \(A(m,n)\) the number of distinct solutions of the congruence \(x^2\equiv n\pmod m\).

MSC:

11F68 Dirichlet series in several complex variables associated to automorphic forms; Weyl group multiple Dirichlet series
11M32 Multiple Dirichlet series and zeta functions and multizeta values
11F37 Forms of half-integer weight; nonholomorphic modular forms

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References:

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