Geodesic
Summability of the Dirichlet series with real exponents for an arbitrary analytic function
Izvestiya. Mathematics , Tome 1 (1967) no. 1, pp. 81-94

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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 $f(z)$. 
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     author = {A. F. Leont'ev},
     title = {Summability of the {Dirichlet} series with real exponents for an arbitrary analytic function},
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A. F. Leont'ev. Summability of the Dirichlet series with real exponents for an arbitrary analytic function. Izvestiya. Mathematics , Tome 1 (1967) no. 1, pp. 81-94. http://geodesic.mathdoc.fr/item/IM2_1967_1_1_a5/