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Sample text

_ 1 qn2 2 ---'C_----'- to our sum, and so on with n3, . .. In this manner, we see that the sum diverges, and CT 2 is proved. Assume CT 2. We shall prove that IX satisfies CT 1 by an argument due to Schanuel. Suppose that a does not satisfy CT 1. Then we can find a sequence of integers qi with Then rjI(qj) > 1/2 jqj for j = 1,2, ... and the sum for rjI converges. This is a contradiction, which proves that a satisfies CT 1. [II, §3] 25 ASYMPTOTIC APPROXIMATIONS We observe that Schanuel's function is very smooth, and behaves as well as possible from the point of view of convexity.

II, §2. NUMBERS OF CONSTANT TYPE There is a special kind of numbers which provides useful examples, and is especially easy to work with. They are characterized by the properties of the next theorem. Theorem 6. The following properties concerning an irrational number IX are equivalent. CT 1. There exists a constant c > 0 such that for all integers q > 0 we have IlqIX11 > c/q. CT 2. For any positive function t/J with convergent sum inequality has only a finite number of solutions. L t/J(q), the 24 ASYMPTOTIC APPROXIMATIONS [II, §2] CT 3.

Has the same discriminant as IX for n ~ 1. If IX is reduced, then IX. is reduced for all n ~ 1. If IX is not necessarily reduced, then IX. is reduced for all n sufficiently large. Proof From the lemma, we conclude at once that IX. has the same discriminant as IX (using the construction of the continued fraction, and Chapter I, §3). This proves (i). Furthermore, IX. > 1. If IX is reduced, and 1 lX=a+{3 with an integer a, and {3 > 1, then -1/{3' = a - IX' > 1, so that {3 is also reduced. This proves (ii).