Asymptotic result

Hi. There's an elegant asymptotic result in analysis as follows.
Let r1, r2... be a discrete sequence of positive reals. Let n1, n2..
be an arbitrary sequence of natural numbers. Let the series
sigma ni/ri^(rho+epsilon) converge for each positive epsilon(Rho also
is positive here) Let T(r) be the function on the positive reals
defined by T(r)=n1+n2_....nk where k is the largest index such that rk
<= r. Then the claim is that T(r) is O(r^(rho+epsilon) as r-> infinity
for each epsilon greater than zero. The converse is also true but
straightforward to prove by asymptotic analysis. Is there a proof of
this result that does not use complex analysis? Proof by complex
analysis is given in (implicit in)Lang's Complex Analysis on page 384
(fourth edition) But obviously a proof by usual asymptotic methods
would be more aesthetically satisfying
.

Relevant Pages

  • Re: Asymptotic result
    ... be an arbitrary sequence of natural numbers. ... sigma ni/ri^converge for each positive epsilon(Rho also ... analysis is given in Lang's Complex Analysis on page 384 ... But obviously a proof by usual asymptotic methods ...
    (sci.math)
  • Re: Asymptotic result
    ... be an arbitrary sequence of natural numbers. ... analysis is given in Lang's Complex Analysis on page 384 ... Is r_i an increasing sequence? ... "Understanding Godel isn't about following his formal proof. ...
    (sci.math)
  • Re: Searching zeros of complex function
    ... > Auslosung der Gleichungen," by E. Schroder, Math. ... > related to Konig's theorem and is implicit in the use of the psuedospectra ... many classic text on Complex Analysis from circa 1950/70 ... (e.g., in Regge Pole analysis). ...
    (comp.lang.fortran)