Functiones et Approximatio Commentarii Mathematici

Remarks on the generalized Lindelöf Hypothesis

J. Brian Conrey and Amit Ghosh

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Within the study of arithmetical Dirichlet series, those that have a functional equation and Euler product are of particular interest. In 1989 Selberg described a class $\mathcal{S}$ of Dirichlet series through a set of four axioms which possibly contain all of these interesting Dirichlet series and made a number of interesting conjectures. In particular, he conjectured the Riemann Hypothesis for this class. We prove that one consequence of the Riemann Hypothesis for functions in $\mathcal{S}$ is the generalized Lindelöf Hypothesis. Moreover, we give an example of a function $D$ which satisfies the first three of Selberg's axioms but fails the Lindelöf Hypothesis in the $Q$ aspect.

Article information

Funct. Approx. Comment. Math., Volume 36 (2006), 71-78.

First available in Project Euclid: 18 December 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

Selberg's class Riemann Hypothesis Lindelöf Hypothesis


Conrey, J. Brian; Ghosh, Amit. Remarks on the generalized Lindelöf Hypothesis. Funct. Approx. Comment. Math. 36 (2006), 71--78. doi:10.7169/facm/1229616442.

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