On the structure of the Selberg class. V: \(1<d<5/3\)
The authors continue their comprehensive study of functions belonging to the Selberg \(\mathcal S\) class of Dirichlet series [see the authors' article for the relevant definitions and properties of \(\mathcal S\), in: Number Theory in Progress, 953--992 (1999; Zbl 0929.11028), see also their works in Acta Math. 182, 207--241 (1999; Zbl 1126.11335), J. Reine Angew. Math. 524, 73--96 (2000; Zbl 0982.11051) and Duke Math. J. 101, 529--554 (2000; Zbl 0997.11072)]. Recall that the extended Selberg class \(\mathcal S^\sharp\) consists of Dirichlet series \(F(s)\) with meromorphic continuation over \(\mathbb C\) such that \((s-1)^mF(s)\) is an entire function of finite order for some integer \(m\geq 0\), which satisfy the functional equation \[ \gamma(s)F(s) = \omega{\bar \gamma}(1-s){\bar F}(1-s), \] where \(| \omega| = 1, {\bar F}(s) = \overline {F({\bar s})}\) and \[ \gamma(s) = Q^s\prod_{j=1}^r\Gamma(\lambda_js+\mu_j),\;Q>0, \,\lambda_j > 0,\,\Re\,\mu_j\geq0. \] If the degree of \(F\in\mathcal S^\sharp\) is defined as \(d_F = 2\sum_{j=1}^r\lambda_j\), then a sensible conjecture asserts that \(d_F\) is always a non-negative integer. The authors make a significant contribution to this subject by proving that there are no functions \(F\) from \(S^\sharp\) satisfying \(1 < d_F < 5/3\). If the order of the pole of \(F(s)\) at \(s=1\) is denoted by \(m_F\), then it is also proved that for \(1 < d < 2\) there exist no \(F\) from \(S^\sharp\) with \(m_F \geq 1\). Furthermore, the authors' arguments show that if the degree conjecture holds for all \(F\in\mathcal S^\sharp\) with \(m_F\geq 1\), then it holds also for all \(F\in\mathcal S^\sharp\). Since it was proved long ago by \textit{H.-E. Richert} [Publ. Inst. Math. Acad. Serbe Sci. 11, 73--124 (1957; Zbl 0082.05802)] that \(\mathcal S^\sharp\) is empty for \(0 < d < 1\), we have now a fairly good knowledge of the structure of \(\mathcal S^\sharp\) for \(d < 5/3\); moreover the authors state that they can extend conditionally their result to the whole range \(1 < d < 2\). For the proof, which is long and deep, the linear twists \[ F(s,\alpha) = \sum_{n=1}^\infty a(n)n^{-s}\exp(-2\pi in\alpha) \quad(\alpha \in \mathbb R) \] are of a fundamental importance, where the \(a(n)\)'s are the coefficients of \(F(s)\). The functions \(F(s,\alpha)\) are continued analytically to the left of the line \(\sigma = \sigma(F)\) for every \(F\in\mathcal S^\sharp\) satisfying \(1 < d_F < 2\) (\(\sigma(F)\) is the abscissa of absolute convergence of \(F(s)\)). The features of such an analytic continuation are used to prove the main result. In the course of the proofs, the incomplete Fox hypergeometric functions \[ H_c(z,s) = {1\over2\pi i}\int\limits_{\Re w=-c}\prod_{j=1}^r {\Gamma(\lambda_j(1-s-w)+\bar{\mu}_j)\over \Gamma(\lambda_j(s+w)+\mu_j)}\Gamma(w)z^w\,\text{ d}w \] also play an important rôle.
scientific article; zbMATH DE number 1965451