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Transcript of Number-Theoretic Functions
is the set of positive integers is said to be a number-theoretic ( or arithmetic) function. Definition 1. Definition 2. If n=p_1^(k_1 ) p_2^(k_2 )…p_r^(k_r ) is the prime factorization of n>1,
then the positive divisors of n are
precisely those integers d of the form
d=p_1^(a_1 ) p_2^(a_2 )…p_r^(a_r ) Theorem 1. If n=12, find
τ(n) and σ(n). Example: Given a positive integer n, let τ(n) denote the number of positive divisors of n and σ(n) denote the sum of these divisors. ∑ f(d)
(d|n) Notation: ∑ f(d)
(d|20) Example: τ(n) = ∑ 1
σ(n)= ∑ f(d)
(d|n) Expressions n=1000 (written) Theorem 2. A number-theoretic function f is
said to be multiplicative if
whenever gcd(m,n)=1. Definition 3. σ(ab)= σ(a) σ(b)
τ(ab)=τ(a)τ(b) Multiplicative Property When n=957, show
that σ(n)=σ(n+1). Example: Verify that [τ(n)][τ(n+1)]=τ(n+3)
holds for n=3655. Try this! Thank