Arithmetic function

In number theory, an arithmetic or number-theoretic function is a special kind of function that helps us study numbers. These functions work with positive whole numbers, like 1, 2, 3, and so on, and their results can be complex numbers, which include real numbers and imaginary numbers.

One common example of an arithmetic function is the divisor function. This function tells us how many numbers can divide a given number evenly. For instance, the number 6 can be divided by 1, 2, 3, and 6, so its divisor function value is 4.

Arithmetic functions can behave in very unpredictable ways, but some of them can be understood better using special mathematical tools, like Ramanujan's sum. These functions are important because they help mathematicians explore the properties and patterns of numbers.

Multiplicative and additive functions

An arithmetic function a is

• completely additive if a(mn) = a(m) + a(n) for all natural numbers m and n;
• completely multiplicative if a(1) = 1 and a(mn) = a(m)a(n) for all natural numbers m and n;

Two whole numbers m and n are called coprime if their greatest common divisor is 1, that is, if there is no prime number that divides both of them.

Then an arithmetic function a is

• additive if a(mn) = a(m) + a(n) for all coprime natural numbers m and n;
• multiplicative if a(1) = 1 and a(mn) = a(m)a(n) for all coprime natural numbers m and n.

Notation

In this article, the symbols ∑ and ∏ are used in special ways. When we write ∑ₚ f(p) or ∏ₚ f(p), it means we are adding or multiplying the values of f(p) for every prime number p. For example, ∑ₚ f(p) = f(2) + f(3) + f(5) + ..., and ∏ₚ f(p) = f(2) × f(3) × f(5) × ...

We can also look at prime powers. The symbols ∑ₚᵏ f(pᵏ) and ∏ₚᵏ f(pᵏ) mean we add or multiply f(pᵏ) for every power of a prime number. For example, ∑ₚᵏ f(pᵏ) = f(2) + f(3) + f(4) + f(5) + f(7) + f(8) + f(9) + ..., and this includes numbers like 4 (which is 2²), 8 (which is 2³), and 9 (which is 3²).

We can also look at divisors of a number. The symbols ∑d|ₙ f(d) and ∏d|ₙ f(d) mean we add or multiply f(d) for every positive divisor d of n, including 1 and n itself. For example, if n = 12, then ∏d|₁₂ f(d) = f(1) × f(2) × f(3) × f(4) × f(6) × f(12).

These ideas can be combined. The symbols ∑ₚ|ₙ f(p) and ∏ₚ|ₙ f(p) mean we add or multiply f(p) for every prime number p that divides n. For example, if n = 18, then ∑ₚ|₁₈ f(p) = f(2) + f(3).

Similarly, ∑ₚᵏ|ₙ f(pᵏ) and ∏ₚᵏ|ₙ f(pᵏ) mean we add or multiply f(pᵏ) for every power of a prime number pᵏ that divides n. For example, if n = 24, then ∏ₚᵏ|₂₄ f(pᵏ) = f(2) × f(3) × f(4) × f(8).

Ω(n), ω(n), ν_p(n) – prime power decomposition

The fundamental theorem of arithmetic tells us that any positive whole number can be written in one way as a product of prime numbers raised to powers. For example, the number 12 can be written as 2² × 3¹.

We can also think of this as an endless product over all prime numbers, where most have a power of zero. The p-adic valuation, written as ν_p(n), shows how many times a prime number p is used in the product for n. If p is used, ν_p(n) is the power; if not, it is zero. This helps us understand the building blocks of numbers.

The functions ω(n) and Ω(n) count these building blocks in different ways. ω(n) counts how many different prime numbers are used, while Ω(n) adds up all the powers used.

Multiplicative functions

σ_k(n), τ(n), d(n) – divisor sums

σ_k(n) is the sum of the _k_th powers of the positive divisors of n, including 1 and n, where k is a complex number.

σ₁(n), the sum of the (positive) divisors of n, is usually denoted by σ(n).

Since a positive number to the zero power is one, σ₀(n) is therefore the number of (positive) divisors of n; it is usually denoted by d(n) or τ(n) (for the German Teiler = divisors).

φ(n) – Euler totient function

φ(n), the Euler totient function, is the number of positive integers not greater than n that are coprime to n.

J_k(n) – Jordan totient function

J_k(n), the Jordan totient function, is the number of k-tuples of positive integers all less than or equal to n that form a coprime (k + 1)-tuple together with n. It is a generalization of Euler's totient, φ(n) = J₁(n).

μ(n) – Möbius function

μ(n), the Möbius function, is important because of the Möbius inversion formula. See § Dirichlet convolution, below.

This implies that μ(1) = 1._

τ(n) – Ramanujan tau function

τ(n), the Ramanujan tau function, is defined by its generating function identity.

Although it is hard to say exactly what "arithmetical property of n" it "expresses", it is included among the arithmetical functions because it is multiplicative and it occurs in identities involving certain σ_k(n) and r_k(n) functions.

c_q(n) – Ramanujan's sum

c_q(n), Ramanujan's sum, is the sum of the _n_th powers of the primitive _q_th roots of unity.

Even though it is defined as a sum of complex numbers, it is an integer. For a fixed value of n it is multiplicative in q:

If q and r are coprime, then c q ( n ) c r ( n ) = c q r ( n ).

ψ(n) – Dedekind psi function

The Dedekind psi function, used in the theory of modular functions, is defined by the formula.

Completely multiplicative functions

λ(n) – Liouville function

The Liouville function, written as λ(n), is a special way to look at numbers. It uses a rule that involves a symbol Ω(n), and the result can be either 1 or -1 depending on the value of Ω(n).

χ(n) – characters

All Dirichlet characters χ(n) are completely multiplicative. Two special characters are important:

The principal character (mod n) is a simple rule that checks if two numbers share common factors. It gives the value 1 if they do not share any factors other than 1, and 0 if they do.

The quadratic character (mod n) uses something called the Jacobi symbol for certain numbers. This character is built from another symbol called the Legendre symbol, which tells us about squares modulo a prime number.

Following the normal convention, when the product is empty, the value is 1.

Additive functions

ω(n) – distinct prime divisors

ω(n) counts how many different prime numbers divide a number n. This count is an example of an additive function. For more details, see the Prime omega function.

Completely additive functions

Some special functions in number theory add up in a very neat way. For example, the function Ω(n) counts how many times prime numbers multiply together to make n, even if a prime is used more than once. This function adds up perfectly when you multiply numbers.

Another function, ν_p(n), tells us how many times a specific prime number p can be divided out of n. This also adds up perfectly when numbers are multiplied.

There’s also something called the logarithmic derivative, which uses these ideas in a special way to connect the arithmetic derivative of a number to its prime factors.

Neither multiplicative nor additive

π(x), Π(x), ϑ(x), ψ(x) – prime-counting functions

These special functions help us understand prime numbers, even though they are not arithmetic functions. They work with real numbers and are used in proofs related to the prime number theorem.

The function π(x) counts how many prime numbers are less than or equal to x. It adds up the characteristic function of prime numbers. Another function, Π(x), counts prime powers, giving each prime its full weight, half weight for its square, and so on.

The Chebyshev functions, ϑ(x) and ψ(x), add up the natural logarithms of primes up to x. The second Chebyshev function ψ(x) adds up the logarithms of prime powers.

Λ(n) – von Mangoldt function

The von Mangoldt function, Λ(n), is zero unless n is a power of a prime number, like p^k. If n is such a power, the function gives the natural logarithm of the prime p.

p(n) – partition function

The partition function, p(n), tells us how many ways we can write n as a sum of positive whole numbers, where the order does not matter.

r_k(n) – sum of k squares

The function r_k(n) counts how many ways n can be written as the sum of k squares. Different orders and different signs of the square roots count as separate ways.

D(n) – Arithmetic derivative

The arithmetic derivative, D(n), works like a regular derivative but for whole numbers. If n is a prime number, D(n) is 1. For multiplying two numbers, it follows a special rule: D(mn) = mD(n) + D(m)n_. This is similar to the product rule.

Summation functions

An arithmetic function is a special kind of math rule that works with whole numbers. One way to study these functions is by adding up their values in a certain way. This added-up version is called a summation function.

We can also use special math tools called generating functions to understand arithmetic functions better. These tools help us see patterns and relationships between different functions. For example, multiplying two generating functions can tell us about a new function created from the original two. This is similar to how combining ingredients in cooking creates a new dish with features from each ingredient.

Relations among the functions

Arithmetic functions are special rules that help us understand numbers better. These rules connect with other mathematical ideas, like powers and logs. There are many formulas that show how these functions relate to each other.

One important idea is called Dirichlet convolution. It helps us combine different arithmetic functions in useful ways. For example, we can use it to find patterns in how numbers break down into smaller parts. Another example is the sum of squares, which tells us how many ways we can write a number as a sum of squares.

These relationships help mathematicians solve problems and discover new patterns in numbers. They also connect to deeper areas of math, like number theory and analysis.

First 100 values of some arithmetic functions