Elementary Number Theory

Front 1

Arithmetic Function

Back 1

Function whose domain is the set of positive integers.

Front 2

Multiplicative arithmetic function

Back 2

An arithmetic function such that f(mn)=f(m)f(n) whenever m and n are relatively prime positive integers.
An arithmetic function is said to be completely multiplicative if f(mn)=f(m)f(n) for all positive integers m and n.

Front 3

Theorem 3.1

Back 3

Let f be an arithmetic function and for n in Z with n>0 let
F(n)= Sum f(d) for d|n, d>0
Then F is multiplicative.

Front 4

Theorem 3.2

Back 4

The euler phi-function is multiplicative.

Front 5

Theorem 3.3

Back 5

Let p be a prime number and let a in Z with a>0. Then phi(p^a)=p^a-p^{a-1}

Front 6

Theorem 3.4

Back 6

phi(n)=n product of (1-1/p) for p|n where p is prime.

Front 7

Theorem 3.5

Back 7

LEt n in Z with n>0 then the Sum of phi(d) for d|n = n

Front 8

Definition 3. v(n)

Back 8

The number of positive divisors function. v(n)=|{d in Z: d>0;d |n|
IE the number of positive divisors of n.

Front 9

Theorem 3.6

Back 9

v(n) is multiplicative.

Front 10

Theorem 3.7

Back 10

Let p be a prime number and let a in Z with a >=0.
v(p^a)=a+1

Front 11

Theorem 3.8

Back 11

Let n=p1^a1*p2^a2*p3^a*...*pr^ar
v(n)= the product of (ai+1) for i in (1 to r)

Front 12

Definition 4. The sum of positive divisors function σ(n)

Back 12

σ(n) = the sum of d for d|n
Also σ(n) is multiplicative
Theorem 3.9

Front 13

Theorem 3.10

Back 13

Let p be a prime number and let a in Z, a>=0
σ(p^a)={p^{a+1}-1}/{p-1}