## Functiones et Approximatio Commentarii Mathematici

### Remarks on the generalized Lindelöf Hypothesis

#### Abstract

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

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

Dates
First available in Project Euclid: 18 December 2008

https://projecteuclid.org/euclid.facm/1229616442

Digital Object Identifier
doi:10.7169/facm/1229616442

Mathematical Reviews number (MathSciNet)
MR2296639

Zentralblatt MATH identifier
1196.11121

#### Citation

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