Abstract: Mainly studies growth of the random Dirichlet series and growth of the H talues random Dirichlet series on the horizontal lines. Discusses the value of rendom Dirichlet series in the Hilbert space. Some important results are obtained：the H values random Dirichlet series ，no matter their convergence domain is the whole plane or half plane ，they a.s. have the same growth on every horizontal line as well as every strip and the whole convergence domain.

Key words: random Dirichlet series, Hilbert space, convergence