Generalisation of the Beurling-Nyman criterion for the Riemann Hypothesis (Q1764182)

The Beurling-Nyman criterion for the Riemann \(\zeta\)-function is generalized to a class of Dirichlet series. Suppose \(F(s)=\sum_{n=1}^{\infty}{a_{n}\over n^s }\), with \(a_{n}\ll n^{\varepsilon}\) for every \(\varepsilon >0\), is a Dirichlet series which is absolutely convergent for \(\Re s > 1\), and which admits a meromorphic continuation to \(\Re s \geq {1\over 2}\) with a unique pole of finite order \(m_{F}\) at \(s=1\). Let \(\Psi_{F}: \mathbb{R}^{+} \to \mathbb{C}\), \[ \Psi_{F}(x) = \text{Res} \biggl({x^{s}\over s}F(s),1\biggr)- \sum_{n\leq x}a_{n}, \] and \(\Psi_{F}^{(1)}(x)=\Psi_{F}(1/x)\). Let \(\widetilde{B_{F}}\) be the space of functions such that \[ f(t)=\sum_{k=1}^{n}c_{k}\Psi_{F} \biggl({\alpha_{k}\over t}\biggr), \] \( c_{k}\in \mathbb{C}\), where \(\alpha_{k}\in (0,1]\) satisfy \(\forall \ell \in [0,m_{F}-1],\, \sum_{k=1}^{n}c_{k}\alpha_{k}(\ln \alpha_{k})^{\ell} = 0\). The main result announced in this paper is as follows. Let \(F\) be such that \(\Psi_{F}^{(1)} \in L^{2}(0,\infty)\). Then the following statements are equivalent: (i) \(F\) does not vanish in \(\Re s > 1/2\); (ii) \(\widetilde{B_{F}}\) is dense in \(L^{2}(0,1)\); (iii) The characteristic function of the interval \((0,1]\) is in the closure of \(\widetilde{B_{F}}\) in \(L^{2}(0,1)\). The main result allows the formulation of similar conditions for the validity of the Generalized Riemann Hypothesis for functions in the Selberg class. The proofs are due to appear in forthcoming articles by the author.