1. We generalize one of Mandelbrojt's theorems Considering Valiron's product we can replace Mandelbrojt's condition C by the following: C~n. Suppose n runs over all the positive integers, and suppose and we can study the representation of F(s) by instead of that of F(s) by In the present case, Mandelbrojt's theorem can be easily modified, and the inequality in the conclusion takes a more convenient form: where2. The Cauchy-Hadamard formula on the radius of convergence of power series can be extended to certain generalized Dirichlet series with complex exponents. Our results are also extensions of those of Hille and,and our form is more convenient. As a consequence of our general theorem, we have, for example, the following result:Theorem If {λ_n}satisfies the condition C′and the following conditions: then the series converges in a half-plane bounded by and it diverges in a half-plane bounded by3. According to Mandelbrojt's method, the preceding results can be used to detect the singularities of a nction represented by some generalized Dirichlet series From our general theorem, we deduce, for example, the following:Theorem Suppose {λ_n}°satisfies C′and (3) and the following conditions:C_1′. For x is sufficiently large, the number of λ_n, whose modulii lie between x and x + 1, is less than a fixed positive number K;C_2~′. Given any δ>0, we have, for n sufficiently large, then each point on the boundry of the half-plane of convergence of is a sigularity of the function represented by the series.4. Applying the inequality (1), we can extend our previous results on the growth of entire functions defined by the Dirichlet series to the case of generalized Dirichlet series whose exponents satisfy the conditions C′, C_1~′ and C_2~′
