# Question

(a) An integer N is to be selected at random from {1, 2, . . . , (10)3} in the sense that each integer has the same probability of being selected. What is the probability that N will be divisible by 3? by 5? by 7? by 15? by 105? How would your answer change if (10)3 is replaced by (10)k as k became larger and larger?

(b) An important function in number theory—one whose properties can be shown to be related to what is probably the most important unsolved problem of mathematics, the Riemann hypothesis—is the Mӧbius function μ(n), defined for all positive integral values n as follows: Factor n into its prime factors. If there is a repeated prime factor, as in 12 = 2 · 2 · 3 or 49 = 7 · 7, then μ(n) is defined to equal 0. Now let N be chosen at random from {1, 2, . . . (10)k}, where k is large. Determine P{μ(N) = 0} as k→∞.

To compute P{μ(N) ≠ 0}, use the identity

where Pi is the ith-smallest prime. (The number 1 is not a prime.)

(b) An important function in number theory—one whose properties can be shown to be related to what is probably the most important unsolved problem of mathematics, the Riemann hypothesis—is the Mӧbius function μ(n), defined for all positive integral values n as follows: Factor n into its prime factors. If there is a repeated prime factor, as in 12 = 2 · 2 · 3 or 49 = 7 · 7, then μ(n) is defined to equal 0. Now let N be chosen at random from {1, 2, . . . (10)k}, where k is large. Determine P{μ(N) = 0} as k→∞.

To compute P{μ(N) ≠ 0}, use the identity

where Pi is the ith-smallest prime. (The number 1 is not a prime.)

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