Arithmetic functions · Definition

Multiplicative arithmetic function

elementary number theoryarithmetic functions

Before you start you need to know

Arithmetic functions assign a number to each positive integer. Multiplicative ones are especially useful because they can be studied prime by prime.

Definition

An arithmetic function \(f\) is multiplicative if \(f(1)=1\) and \(f(mn)=f(m)f(n)\) whenever \(\gcd(m,n)=1\). Its behavior is determined by its values on prime powers.

\[f(mn)=f(m)f(n)\text{ if }\gcd(m,n)=1\]

If \(f\) is multiplicative, then \(f(n)=\prod f(p^{a})\) over the prime factorization \(n=\prod p^{a}\), by the fundamental theorem of arithmetic. Moreover, \(f\) is fixed by its values on prime powers. For example, the divisor function \(\tau(n)\) satisfies \(\tau(p^{a})=a+1\).

For \(n=360=2^3\cdot 3^2\cdot 5\), we get \(\tau(360)=(3+1)(2+1)(1+1)=24\), matching a direct count of divisors. This shows how the computation reduces to independent prime factors.

A typical mistake is assuming \(f(m+n)=f(m)+f(n)\) or that \(f(mn)=f(m)f(n)\) holds without the condition \(\gcd(m,n)=1\). For instance, \(\tau(4)=3\) but \(\tau(2)\tau(2)=4\), so the rule fails without coprimality.

Think about it

What condition is needed for \(f(mn)=f(m)f(n)\) to hold in the definition?

\(\gcd(m,n)=1\)

Exercises 10

  1. What is the value of \(\tau(12)\)?
  2. If \(f\) is multiplicative and \(\gcd(m,n)=1\), which expression equals \(f(mn)\)?
  3. What is the value of \(\tau(p^{3})\) for a prime \(p\)?
  4. Which of these functions is not multiplicative?
  5. If \(f\) is multiplicative, what is \(f(1)\)?
  6. If \(\tau(n)=12\) and \(n=2^{2}\cdot 3^{?}\), what is the exponent of 3?
  7. Compute \(\tau(2^{4}\cdot 5^{2})\).
  8. If \(f\) is multiplicative with \(f(2)=3\) and \(f(5)=4\), what is \(f(10)\)?
  9. When computing \(\tau(4)\) as \(\tau(2)\cdot\tau(2)\), why does the multiplicativity rule not apply?
  10. Let \(f\) be multiplicative with \(f(p^{a})=a+1\) for every prime \(p\). What is \(f(360)\)?

References

  • David M. Burton, Elementary Number Theory, 7th edition, McGraw-Hill, 2011. cap. 6
  • Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery, An Introduction to the Theory of Numbers, 5th edition, Wiley, 1991. cap. 4