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Dirichlet series; Banach algebra; topological zero divisor; division algebra; continuous linear functional; total set
Let $F$ be a class of entire functions represented by Dirichlet series with complex frequencies $\sum a_k {\rm e}^{\langle \lambda ^k, z\rangle }$ for which $(|\lambda ^k|/{\rm e})^{|\lambda ^k|} k!|a_k|$ 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.
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