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Analytic number theory: an introductory course by P. T. Bateman, Harold G. Diamond

By P. T. Bateman, Harold G. Diamond

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17 each mapping is continuous. 1 The 1 function as an exponential Since 1 E A1, it is representable as an exponential. 20. We work with p, the inverse function of 1, for simplicity in calculations. For p a prime, let p p = e - e p . By the power series for log, we have Thus, 1, = exp fp, where fp is the sum of the last series. By the homomorphism property of exp, for any prime P , we have The left side of the last equation converges to 1 as P + 00 (cf. 20). If we define K as fi + f3 + f5 - - - , then + K(n) := some p and some j 2 1, l/j, if n = 9, if n = 1 or n is divisible by more than one prime.

F5)(2). a 03 the special role that . {fi(l)} plays? 12 Let { fi)gl be a sequence of arithmetic functions, none of which is identically zero. Also assume that f l * f 2 * - - - converges. Prove that f l * f 2 * # 0. Hint. Consider the f; for which fi(1) = 0. 11 Let {hj}j”,l be a sequence of arithmetic functions with h j ( 1 ) = 0 f o r all j . Then, for all indices u > logn/log2, (hl * Proof. Suppose u - . * h,)(n) = 0. > logn/log2. In at least one ni satisfies ni 5 n1lu < 2. Thus ni = 1, hi(ni) = 0 , and so (hl * .

4) that y satisfies y = lim ( C - - 1 logx). X+OO n nsx If 0 < a < 1, we take f ( t ) = t-" and apply Euler's formula as in the case a = 1. 6). If a > 1, we write n 1 are 0 uniform in a (as well as in x, of course). 5) cannot be changed t o o(x-"). Euler's constant turns up in various areas of mathematics; it is a famous unsolved problem whether y is rational or irrational.

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