Math 115. Slides from the Lecture of November 19

This handout contains the slides from the lecture of November 19.

Arithmetic Functions

Definition.

(a). An arithmetic function is a function from ℤ_{>0} to ℂ. (b). A multiplicative function is an arithmetic function f which is not the zero

function, and which satisfies f(mn) = f(m)f(n) for all relatively prime m, n ∈ ℤ_{>0} .

(c). A totally multiplicative function is an arithmetic function f which is not the zero function, and which satisfies f(mn) = f(m)f(n) for all m, n ∈ ℤ_{>0} .

Examples.

• The constant function 1 is totally multiplicative, and multiplicative

• So is f(n) = n^{r}, for any r ∈ ℝ

• If f is totally multiplicative, then it is multiplicative

• The Euler φ-function is multiplicative, but not totally multiplicative

Examples.

• The function f(m) equal to the number of quadratic residues modulo m is multiplicative but not totally multiplicative

• The function f(m) equal to the number of primitive roots modulo m is not multiplicative (why?)

Notation: When writing ∑_{d|n} or ∏_{d|n}, we’re tacitly requiring d > 0 .

Definition. For n ∈ ℤ_{>0} , define: • d(n) = the number of positive divisors of n: ∑_{d|n} 1

• σ(n) = the sum of the positive divisors of n: ∑_{d|n} d

• σ_{k}(n) = ∑_{d|n} d^{k} ( σ_{0}(n) = d(n) , σ_{1}(n) = σ(n) )

• ω(n) = the number of primes dividing n

• Ω(n) = the number of primes dividing n, with multiplicity

Note that

Ω(∏ p^{α(p)}) = ∑ α(p)

Also d(∏ p^{α(p)}) = ∏(α(p) + 1) . This is multiplicative.

And, e^{ω(n)} and e^{Ω(n)} are multiplicative and strongly multiplicative, respectively.

A multiplicative function is uniquely determined by its values on prime powers (as for ϕ(m) ).

A totally multiplicative function is uniquely determined by its values on prime numbers.