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A conjectural extension of Hecke’s converse theorem. (English) Zbl 1443.11034

In this paper, the authors give five proofs to the Conjecture 1.1 which is:
Conjecture 1.1. Let \(\xi \) be a Dirichlet character modulo \(N\), \(k\) a positive integer satisfying \(\xi (-1)=(-1)^{k}\) , and \(\{f_{n}\}_{n=1}^{\infty}\), \(\{g_{n}\}_{n=1}^{\infty}\) sequences of complex numbers satisfying \(f_{n},g_{n}=O(n^{\sigma})\) for some \(\sigma >0\). For \(q\in \mathbb{N}\), let \[ c_{q}(n)=\sum_{a(\text{mod}\ q),(a,q)=1}e(\frac{an}{q}) \] be the associated Ramanujan sum, where \(e(x):=e^{2\pi ix}\), and define \[ \Lambda_{f}(s,c_{q})=\Gamma _{\mathbb{C}}(s+\frac{k-1}{2})\sum_{n=1}^{\infty}\frac{f_{n}c_{q}(n)}{n^{s+\frac{k-1}{2}}} \] and \[ \Lambda_{g}(s,c_{q})=\Gamma _{\mathbb{C}}(s+\frac{k-1}{2})\sum_{n=1}^{\infty}\frac{g_{n}c_{q}(n)}{n^{s+\frac{k-1}{2}}} \] for \(\operatorname{Re} (s)>\sigma +1-\frac{k-1}{2}\). For every \(q\) coprime to \(N\), suppose that \(\Lambda_{f}(s,c_{q})\) and \(\Lambda_{g}(s,c_{q})\) continue to entire functions of finite order and satisfy the functional equation \[ \Lambda_{f}(s,c_{q})=i^{k}\xi (q)(Nq^{2})^{\frac{1}{2}-s}\Lambda_{g}(1-s,c_{q}). \] Then \(f(z):=\sum_{n=1}^{\infty}f_{n}e(nz)\) is an element of \(M_{k}(\Gamma_{0}(N),\xi ).\)

MSC:

11F11 Holomorphic modular forms of integral weight
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F06 Structure of modular groups and generalizations; arithmetic groups

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References:

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