Whenever you use a result or claim a statement, provide a justification or a proof, unless it has been covered in the class. In the later case, provide a cita- tion (such as“by the 2- out- of-3 principle”or“by Coro. 0.31 in the textbook”).

You are encouraged to discuss the problems with your peers. However, you must write the homework by  yourself using your words and acknowledge your collaborators.

Problem 1. For this problem, you may want to review one-variable Calculus

(a)  (3 pts) Recall the definition (In this course, log = loge denotes the natural logarithm)

Li(x) := \2 北           (x > 2).

Question: What is the d北(d)Li(x) of Li(x)?

(b)  (5 pts) Two real functions f(x) and g(x) are asymptotically equal if

f(x)

北→∞ g(x)

Prove that: Li(x) and log(北) 北  are asymptotically equal.

Problem 2 (5 pts). Let p be a prime number and k,l be two natural numbers. Show that

k                         l

之 σi(pl) =之 σi(pk ).

i=0                     i=0

Problem 3 (5 pts). Let n be a positive integer and k a natural number. Show that

σk(n) = σ−k(n)nk .

Conclude that n is perfect if and only if σ−1(n) = 2.

Problem 4. We say that a positive integer n is square-free if n is not divisible by p2  for any prime number p. (E.g. 15 and 37 are square-free, but 24 and 49 are not.) Consider the arithmetic function µ (named after A.F. Möbius, popularly known for his strip) as follows:

「 1          if n = 1,

µ(n) :=〈( sq(N)a(O)

(a)  (3 pts) Compute µ(n) for n = 1, ··· , 15.