Summability of the Dirichlet series with real exponents for an arbitrary analytic function

Given a function $f(z)$ that is analytic on a closed vertical interval of length $2\pi\sigma$, we use a definite rule to associate with it a formal Dirichlet series with exponents $\pm\lambda_n(n=1,2,\dots)$, where $\lambda_n>0$ and $\lim\limits_{n\to\infty}\frac{n}{\lambda_n}=\sigma$. In general this series diverges everywhere. We give a method for summing it to the function $f(z)$.