Arithmetic Functions Overview

Number Theory · Axiom Academy

LESSON Arithmetic Functions Overview Introduction to functions from positive integers to complex numbers and their multiplicative properties 1. What is an Arithmetic Function? An arithmetic function (or number-theoretic function) is any function whose domain is the positive integers and whose range is a subset of the complex numbers. Typically, arithmetic functions take real or integer values and encode information about the number-theoretic properties of their inputs, such as divisibility, prime structure, or multiplicative relationships. Let's explore four of the most important arithmetic functions in number theory: 3. Computing Examples for n = 12 Let's calculate each function's value at n = 12 to see them in action. First, note that 12 = 2 2 × 3 and its divisors are 1, 2, 3, 4, 6, 12 . 4. The Power of Multiplicativity An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m) · f(n) whenever gcd(m, n) = 1. All four of our example functions (d, σ, φ, μ) are multiplicative! This means we can compute their values on any number by: For multiplicative functions, we only need formulas for prime powers p k . Here are the formulas for our four key functions:

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