Abstract

this paper we will first prove the theorem above, then extend it to a form from which one can find as many terms of the expansion as desired, and finally discuss the relationship of our work to earlier results of Moser and Wyman [3]. * Research supported by the United States Office of Naval Research. 2. Proof of theorem 1 We will write i n (s) for P n j=1 j \Gammas , which, if s ? 1, is the nth partial sum of the Dirichlet series for the Riemann zeta function. The following computation will be the basis for the developments in the sequel. We have, from the familiar generating function, = (1 + x)(1 + ) \Delta \Delta \Delta (1 + log(1 + )g sj exp f(\Gamma1) g: (1) In the last member we wish to replace all of the i n 's by their asymptotic expansions in powers of n and log n. These expansions are (e.g., [5], p. 124) i n (s) log n + fl + d(1; j)n \Gammaj ; if s = 1; i(s) + (1 \Gamma s) \Gammas+1 j=1 d(s; j)n \Gammas\Gammaj+1 ; if s ? 1, (2) where d(s; j) = \Gamma B j s + j \Gamma 2 and the B j 's are the Bernoulli numbers. By combining (1) and (2), we now have exp x(log n + fl + d(1; j) \Theta i(s) + 1 (1 \Gamma s)n d(s; j) s+j \Gamma1 (3) in the sense that for each fixed k, the coefficient of x on the left has, for its complete asymptotic series as n !1, the coefficient of x on the right. Now we can prove theorem 1. To do that we need only keep the portion of the development in powers of n and log n that does not approach zero on the right side of (3), along with the size of the first neglected term. If we take the coefficient of x on both sides of (3) we obtain x(log n+fl) \Psi\Psi The second exponential factor above is ...

Citations

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