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)
(1) Let f be a multiplicative function satisfying Σ f(d)-n/0(n), where the sum is taken over all positive divisors of n, and ф is Euler's totient function. Use the Mobius inversion formula to prove that f(n) ."(n)/0(n) (1) Let f be a multiplicative function satisfying Σ f(d)-n/0(n), where the sum is taken over all positive divisors of n, and ф is Euler's totient function. Use the Mobius inversion formula to prove that f(n) ."(n)/0(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...
oblemns for Solution: 1. Recall that Euler's phi function (or called Euler's totient function) φ(n) is defined as the number of integers m in the range 1< m<n such that m and n are relatively prime, i.e., gcd(rn, n) l. Find a formula for φ(n), n 2. (Hint: Factor n as the product of prime powers. i.e., n-TiỀ, where pi's are distinct primes and ei 〉 1, i, where p;'s are distinct primes and e > 1 t. oblemns 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...
Exercise 2. Let φ denote the Euler totient function. (i) Prove that for all positive integers m and n, if m,n are relatively prime (coprime), then φ(mn-o(m)o(n) (ii) Is the converse true? Prove or provide a counter-example.
Prove that if f is a multiplicative arithmetic function then f([m, n])f((m, n)) = f(m)f(n) for all positive integers m and n. Hint: [m, n] is the least common multiple of m and n and (m, n) is the greatest common divisor of m and n.
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...
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