Absolutely convergent dirichlet series and analytic continuation of its sum.

(Lobachevskii Journal of Mathematics, Volume I)

The paper contains some results on analytic continuation of the sum of Dirichlet series obtained with the help of the wellknown Ploya theorem. A special attention is paid to an effective determination of the domain into which the sum of series can be continued analytically. Some methods of the effective continuation of the sum of Dirichlet series are considered including, in particular, the analytic continuation by means of initial series. In this part of paper the author employs the results of Leont'ev and other russian mathematicians including his own. A many dimensional analogue of Polya theorem is also obtained as well as some results on analytic continuation of its sum. Finally, the characterization of the exact domain of absolute convergence of many-dimensional Dirichlet series is given under comparatively mild restriction.