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On the structure of the Selberg class. II: Invariants and conjectures. (English) Zbl 0982.11051
The authors continue (the first paper was published in [Acta Math. 182, 207-241 (1999; Zbl 1126.11335)]) their interesting research on the well-known axiomatic Selberg class \(\mathcal S\) of Dirichlet series. Let \(\mathcal S^{\#}\) denote a larger class where Ramanujan’s hypothesis and Euler’s product are dropped. A \(\gamma\)-factor of a function \(F\in\mathcal S^{\#}\) is an expression of the form \(\gamma(s)=Q^s \prod_{j=1}^r \Gamma(\lambda_j s+\mu_j)\), with \(Q>0\), \(\lambda_j>0\) and \(\operatorname{Re} \mu_j\geq 0\) for \(j=1,...,r\), such that \(\Phi(s)=\gamma(s)F(s)\) satisfies a functional equation of the type \(\Phi(s)= \omega \bar{\Phi}(1-s)\) where \(|\omega|=1\). It is known that the functional equation satisfied by a function \(F\in\mathcal S^{\#}\) with degree \(d_F>0\) is not unique in the sense that the data \(Q\), \(r\), \(\lambda_j\), \(\mu_j\) and \(\omega\) are not uniquely determined by \(F(s)\).
In Theorem 1, the authors give an explicit transformation algorithm for any two \(\gamma\)-factors of \(F\). For Dirichlet \(L\)-functions, all \(\gamma\)-factors are given in Proposition 2.1.
Then the concept of an invariant of \(\mathcal S^{\#}\) is defined. Roughly speaking, the invariant is an expression depending on the data \(Q\), \(r\), \(\lambda_j\), \(\mu_j\), \(\omega\) which remains stable for any data corresponding to the same function \(F\). Theorem 1 enables the invariants of functions \(F\in\mathcal S^{\#}\) to be characterized, which is done in Theorem 2.
After that the authors propose several new and known conjectures that illustrate the relevance of such invariants to the structure of the Selberg class \(\mathcal S\). The last two theorems are devoted to relationships between these conjectures.

MSC:
11M41 Other Dirichlet series and zeta functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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[1] Conrey J. B., Duke Math. J. 72 pp 673– (1993)
[2] Kaczorowski J., Acta Math. 182 pp 207– (1999)
[3] M. R. Murty, Selberg’s conjectures and Artin L-functions, Bull. A. M. S. 31 (1994), 1-14. · Zbl 0805.11062
[4] A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Proc. Amal Conf. Analytic Number Theory, ed. by E. Bombieri et al., UniversitaA di Salerno (1992), 367-385; Collected Papers, vol. II, Springer Verlag (1991), 47-63.
[5] Springer Lect. Notes Math. 627 pp 79– (1977)
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