C5. Let n EZ. If f is a multiplicative arithmetic function and pi is the prime factorization of 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...
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
(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)
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)
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.
Consider the von Mangoldt function, defined as follows. if n is a power of the prime p otherwise, logp A(n)- 0 where logp denotes the natural logarithm of p. For example, A(8) A(12) 0, A(67) log 67. Show that log 2, logn -XA(d) dln Deduce that A(n) Σμ(d) log (7) 72 where μ denotes the Möbius function.
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...
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer d there exist unique integers q and r such that n = dq + r and 0 r< d. We define the mod function as follows: (, r r>n = qd+r^0<r< d) Vn,d E Z d0 Z n mod d That is, n mod d is the remainder of n after division by d (a) Translate the following statement into predicate logic:...
Please Answer 135 Below Completely: Definition Let E-R and f : E-+ R be a function. For some p E E' we say that f is continuous at p if for any ε > 0, there exists a δ > 0 (which depends on ε) such that for any x E E with |x-Pl < δ we have If(x) -f(p)le KE. This is often called the rigorous δ-ε definition of continuity. A couple of things to note about this definition....
(a) Suppose that f is continuous on [0, 1] and f(o) = f(1). Let n be 20. any natural number. Prove that there is some number x such that f fx+1/m), as shown in Figure 16 for n 4. Hint: Consider the function g(x) = f(x)-f(x + 1/n); what would be true if g(x)ヂ0 for all x? "(b) Suppose 0 < a 1, but that a is not equal to 1/n for any natural number n. Find a function f...