n) is a multiplicative function. Show that the function Define a (c) n multiplicative function. n)...
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z[i] (b) Prove that if a + r є z[i] is a unit of Zli], then Ma + bi)-1. (c) Find all of the units in Zli 5. Let Zli_ {a + bi l a,b...
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...
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
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)
5. Suppose n > 0 Show that if ā is the (multiplicative) inverse of a modulo n then erpn(а)-erph (a). (Hint. Consider ākak-Ga)k-1k-1 (mod n)) 5. Suppose n > 0 Show that if ā is the (multiplicative) inverse of a modulo n then erpn(а)-erph (a). (Hint. Consider ākak-Ga)k-1k-1 (mod n))
Let A be n × n with AT-A. (The matrix A is syrnmetric.) Let B be 1 × n and let c E R. Define f : Rn → R by f(x) = 2.7, A . x + B . x + c. Show that The function f is a quadratic function Let A be n × n with AT-A. (The matrix A is syrnmetric.) Let B be 1 × n and let c E R. Define f : Rn...
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
(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)
Problem 3 (a) Use the definitional method to find the smallest big-O estimate for the following following function. You must determine the multiplicative constant C and the threshold constant k the multiplicative constant C and ihet Show calculation. (b) Use the limit method to formally determine the asymptotic relationship between the following two functions. Show the steps of your calculation and clearly state each rule that you used in the calculation. fn) 100log2n and g(n)-log1on [4+ 4 points] Problem 3...
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.