Abstract. We prove a Tauberian theorem concerning the summability method $D_{\lambda,a}$ based on the Dirichlet series $\sum a_ne^{-\lambda_nx}$ with $a_{n+1}\sim a_n>0$ when the sequence $(\lambda_n)$ satisfies the `high indices' condition $\lambda_{n+1}>c\lambda_n\ge0$ with any $c>1.$

AMS Subject Classification (1991): 40E05, 40G99

Keyword(s): Tauberian, Dirichlet series methods, high indices

Received June 20, 1997. (Registered under 2652/2009.)