Question

We begin by formally defining the arithmetic function v(n) first introduced irn Example 1(b). Definition 3: Let neZ with n > 0. The number of positive divisors function, denoted v(n), is the function defined by In other words, v(n) is the number of positive divisors of n. [The notation here is chosen by this author for ease of remembrance: v (lowercase Greek letter nu) represents the "number" of positive divisors. However, the number of positive divisors function is denoted variously by d(n) and t(n) .] A2. Let n-P1 P2 Prn, where P1, P2, , Prn are distinct primes. Prove that v(n-2". For convenience, here's a summary of some potentially useful definitions and facts from our last lecture: For any two arithmetic functions f and g, the convolution of f with g is f(n) * g(n) = (f * g)(n) = dn, d 0 d n, d>0 1 denotes the constant function which maps every n to 1 (i.e. 1(n)-1 for all nEZ+) The convolutive identity function is δ(n) = 0, n>1 The Möbius μ-function is μ(n)- (-1), n-p1P2 Pr with P1, P2, , Pr being distinct primes 0, p2n for some prime p For any arithmetic f and g: f*g-g*f, f*(9 * h)-(f * g) * h, f*δ-δ*f-f, μ*1-1*μ-δ The Möbius Inversion Formula (Theorem 3.15) says: For any arithmetic f and g, d n, d 0 which is equivalent to: fg1

Answer 1

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