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C) Prove that (log log x)k = o((log x)" ) for every " > 0 and every k > 0. 1 1 X + + (d) Find with proof a function f : ! such that x log x = o(f (x)) and n=1 f (n) diverges. (2) Let pn denote the nth prime. It is known that pn cn log n for some constant c. Using this information and Theorem 31, prove that c = 1. R R The Number of Prime Divisors of n: Notation. (n). De nition. Let f : + ! + and g : + ! + . Then f (n) is said to have normal order g(n) if for every " > 0, the number of positive integers n x satisfying (1 ?

K xk X1 (c) Prove that = log x + O(1). k k x (3) (a) How many positive integers 210 are not divisible by each of the primes 2, 3, 5, and 7? For example, 11 would be such an integer but 39 would not be. (b) Let A(x) = jfn x : each of 2; 3; 5; and 7 does not divide ngj. Prove that A(x) cx for some constant c and determine the value of c. (4) Let a be a real number. Suppose f : a; 1) ! has the property that for every t a, there exists an M (t) such that jf (x)j M (t) for all x 2 a; t]. Suppose g : a; 1) !

De ne 1 X T (x) = jfp x : p 2 T gj. Suppose that p1 converges. , that almost all primes are not in T )? Riemann-Stieltjes Integrals: De nitions and Notations. Suppose f : a; b] 7! 1 < xn = b. x g. 1 ; xk ] for k 2 f1; 2; : : : ; ng. Consider R S (P ; f; ftk g) = n X k=1 f (tk )(xk ? 1 ): If S (P ; f; ftk g) tends to a limit A (independent of the tk ) as M (P ) tends to zero, then we write Z b f (x) dx = A a and say that the Riemann integral of f (x) on a; b] exists and equals A. Let g : a; b] 7!