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# Number-Theoretic Functions

Charie N. Magware

by

Tweet## charie magware

on 8 October 2012#### Transcript of Number-Theoretic Functions

Number-Theoretic Functions Charie N. Magware Any function whose domain of definition

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

(d|n)

and

σ(n)= ∑ f(d)

(d|n) Expressions n=1000 (written) Theorem 2. A number-theoretic function f is

said to be multiplicative if

f(mn)=f(m)f(n)

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

YOU! Example:

Full transcriptis 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

(d|n)

and

σ(n)= ∑ f(d)

(d|n) Expressions n=1000 (written) Theorem 2. A number-theoretic function f is

said to be multiplicative if

f(mn)=f(m)f(n)

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

YOU! Example: