zbMATH — the first resource for mathematics

On the Selberg class of Dirichlet series: Small degrees. (English) Zbl 0796.11037
Atle Selberg introduced a class of Dirichlet series \(F(s)= \sum_{n=1}^ \infty a_ n n^{-s}\) satisfying the following conditions: (1) \((s-1)^ m F(s)\) is an entire function of finite order for some nonnegative integer \(m\), (2) \(a_ n\ll n^ \varepsilon\), (3) a functional equation of the Riemann type holds involving a “gamma factor” of the type \(\prod_{i=1}^ k \Gamma(w_ i s+\mu_ i)\) with \(w_ i>0\) and \(\text{Re } \mu_ i\geq 0\), (4) \(F(s)\) has an Euler product. It is reasonable to conjecture that Riemann’s hypothesis holds for this class of functions. In addition, Selberg made some other conjectures whose consequences are discussed in the present paper. However, the main topic is the quantity \(d_ F= 2\sum_{i=1}^ k w_ i\) called the degree of \(F\). For instance, \(d_ \zeta=1\) for Riemann’s zeta-function. It is shown that the degree 1 is actually minimal if the constant function \(F=1\) is excluded. It follows that any function in the Selberg class can be factored into a product of “primitive” functions; these are functions which cannot be written as a product of nontrivial functions of the same class. Also, \(F\) is primitive if \(d_ F<2\). The last two sections of the paper are devoted to the important case \(w_ i=1/2\) when the degree is 1 or 2; in all known cases, the gamma factor can be written so that indeed \(w_ i=1/2\).
Reviewer: M.Jutila (Turku)

11M41 Other Dirichlet series and zeta functions
Full Text: DOI
[1] S. Bochner, On Riemann’s functional equation with multiple Gamma factors , Ann. of Math. (2) 67 (1958), 29-41. JSTOR: · Zbl 0082.29002 · doi:10.2307/1969923 · links.jstor.org
[2] C. Epstein, J. Hafner, and P. Sarnak, Zeros of \(L\)-functions attached to Maass forms , Math. Z. 190 (1985), no. 1, 113-128. · Zbl 0565.10026 · doi:10.1007/BF01159169 · eudml:173615
[3] P. Gerardin and W. Li, Functional equations and periodic sequences , Théorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 267-279. · Zbl 0688.10039
[4] M. I. Gurevich, Determining \(L\)-series from their functional equations , Math. USSR-Sb. 14 (1971), 537-553. · Zbl 0241.12006 · doi:10.1070/SM1971v014n04ABEH002819
[5] E. Hecke, Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung , Math. Ann. 112 (1936), 664-699. · Zbl 0014.01601 · doi:10.1007/BF01565437 · eudml:159847
[6] H. Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen , Math. Ann. 121 (1949), 141-183. · Zbl 0033.11702 · doi:10.1007/BF01329622 · eudml:160177
[7] A. Selberg, Old and new conjectures and results about a class of Dirichlet series , in Collected Papers, Volume 2, Springer-Verlag, Berlin, 1991, pp. 47-63.
[8] C. L. Siegel, Bemerkungen zu einem Satz von Hamburger über die Funktionalgleichung der Riemannsche Zetafunktion , Math. Ann. 86 (1922), 276-279. · JFM 48.1216.01
[9] E. C. Titchmarsh, The theory of the Riemann zeta-function,second edition , The Clarendon Press Oxford University Press, New York, 1986. · Zbl 0601.10026
[10] M.-F. Vignéras, Facteurs gamma et équations fonctionnelles , Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Springer-Verlag, Berlin, 1977, 79-103. Lecture Notes in Math., Vol. 627. · Zbl 0373.10027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.