A study of various results for a class of entire Dirichlet series with complex frequencies. (English) Zbl 1463.30011

Summary: Let \(F\) be a class of entire functions represented by Dirichlet series with complex frequencies \(\sum a_k\text{e}^{\langle\lambda ^k,z\rangle}\) for which \((|\lambda ^k|/\text{e})^{|\lambda ^k|}k!|a_k|\) is bounded. Then \(F\) is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. \(F\) is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to \(F\) have also been established.


30B50 Dirichlet series, exponential series and other series in one complex variable
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
17A35 Nonassociative division algebras
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