(1) Let f be a multiplicative function satisfying Σ f(d)-n/0(n), where the sum is taken over all ...
Let f be a multiplicative function satisfying ∑f(d) = n/φ(n), where the sum is taken over all positive divisors of n, and φ is Euler's totient function. Use the Mobius inversion formula to prove that f(n)=μ2(n)/φ(n)
for each positive integer m, let v(m) denote the number of divisors of m. define the function F(n) =∑ v (d) dIn where the sum is over all positive divisors d of n prove that function F(n) is multiplicative
Lambda(n) denotes the Louisville function (ie the completely multiplicative function satisfying lambda(p): = -1 for every prime number p) Let f be a completely multiplicative function that is not identically zero and suppose that the series F) 28 converges absolutely for Re(s) > ơa. Prove that F(s) 0 and that f(n)x(n) _ F(22) (Re(s) > σα). 28 仁! Let f be a completely multiplicative function that is not identically zero and suppose that the series F) 28 converges absolutely for...
C5. Let n EZ. If f is a multiplicative arithmetic function and pi is the prime factorization of n. prove that μ(d)/(d)-| | (1-f(pi)) d n, d>0 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...
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...
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0 for all x ∈ (0,∞). (a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(k + 1) for all k ∈ N. (b) Use (a), show that Xn−1 k=1 f '(k) ≤ f(n) − f(1) ≤ Xn k=2 f '(k). (c) Let r > 1. By finding...
5.7. Let n an E C be a multiplicative function defined by a 1 and ifn-2, pθ and 0 < θ < 1 . Prove that, as x → oo, we have where lEpl ano(a) for some constant r. 5.7. Let n an E C be a multiplicative function defined by a 1 and ifn-2, pθ and 0
4) Let D be the set of all finite subsets of positive integers. Define a function (:2 - D as follows: For each positive integer n, f(n) =the set of positive divisors of n. Find the following f (1), f(17) and f(18). Is f one-to-one? Prove or give a counterexample.
T(n) is the number of divisors of n, and u(n)-1 Define an arithmetic function A as follows: if p is a prime and k 1 let A(p) log p for all other n, let A(n) 0. (Warning: A is NOT a multiplicative function!) Prove that (A* u)(n) log n for all n. (HINT: consider the various d which divide n expressed in terms of the prime factorization of n
4 Suppose f : (0,0) → (0,x), is a differentiable function satisfying f(a +b)-f(a)fb), for all a,b>0 Moreover, assume that f(0)1 (a) Prove that there exists λ (not necessarily positive) such that f(r) = e-Ar, for all r. Hint Find and solve a proper differential equation. (b) Suppose that X is a continuous random variable, with P(X>ab)-P(>a)P(X> b), for all a, b e (0, oo). Prove that X is exponentially distributed