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It is easy to see that p(a) = 0 for a > n o . 2. For example, it is continuous, nonnegative, non-increasing. Its actual determination is usually very difficult. 2 is best possible in a certain sense. 9. Inversion The inversion problem for a Dirichlet series expansion is the determination of the coefficients in terms of the expanded function. 9. 1) and of radius less than one. 2) becomes the path of integration now being part of a vertical line. When the periodicity of F(e-s) is abandoned, as it must be for the sumf(s) of a general Dirichlet series, the factor (1 - e-S)-l may be replaced by s-l (having the same residue at s = 0), and the path of integration may be taken as a whole vertical line.

1 and the familiar uniqueness property for power series. 48 2. Dirichlet Series 12. Summary By way of summarizing the more important results of the present chapter let us list them briefly here in juxtaposition with the corresponding ones for power series. m OD 1. Convergence I JzI < p * > r c m 2. Differentiation F’(z) = m 1 kakzk-’ f’(s) = - k= 0 3. Analyticity IzI < p , 4. Uniqueness F= 5. Inversion c O* c7 o*ak =0 1 > fJC f = O=ak = 0 c oc, c > 0 I ak > 03singularity at s = rc We also list here two important points where analogy with power series fails.

Here we have used the assumed rate of growth of Un and the fact that for a > 0 exp Aka < exp ta, ;Ik c t. 1) converges for D > a, the proof is complete. The following result is a partial converse to the above. 2. m 1. C a, exp( - a) converges, some a > 0 k= 1 n 3 Un = C ak = O(exp a&), k= 1 n + 00. 6. ,,a), n + a. This completes the proof. Notice where it would fail if a c 0. We now prove Cahen's formula. 1. ,s) k= 1 is a (or +a). 2) converges 0 > Q. If E > 0, then hypothesis 1 implies that n C ak= o[exp(a+ k= 1 E)A,], n --t co.

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