Arithmetic function

An arithmetic function is a function whose domain is the positive integers, usually taking values in the complex numbers.

Multiplicative arithmetic function

An arithmetic function

f

is multiplicative when

f (mn) = f (m) f (n)

whenever

m

and

n

Completely multiplicative function

An arithmetic function

f

is completely multiplicative when

f (mn) = f (m) f (n)

for every pair of positive integers

m, n

, without a coprimality condition.

Divisor sum

The divisor sum of an arithmetic function

f

is

F (n) = \sum_{d ∣ n} f (d) .

If

f

is multiplicative, then

F

is multiplicative as well.

Pretentious distance ( $D (f, g; x)$ )

For multiplicative functions bounded by one, the pretentious distance up to

x

is defined by

D (f, g; x)^{2} = \sum_{p \leq x} \frac{1 - ℜ ( f ( p ) g ( p ) )}{p} .

It measures how similarly

f

and

g

behave on primes.

Triangle inequality for pretentious distance

The pretentious distance obeys

D (f_{1} f_{2}, g_{1} g_{2}; x) \leq D (f_{1}, g_{1}; x) + D (f_{2}, g_{2}; x) .

Archimedean character ( $n^{i t}$ )

An Archimedean character on the positive integers is the completely multiplicative function

n \mapsto n^{i t}

for a real parameter

t

.

Halász theorem

Halász's theorem bounds the mean of a bounded multiplicative arithmetic function in terms of its least pretentious distance from an Archimedean character. A large mean can occur only when this distance is small.

Strong aperiodicity of the Möbius function

The Möbius function stays far in pretentious distance from every Archimedean character. One useful uniform form is

in f_{∣ t ∣ \leq 2 (l o g x)^{1/10}} D (μ, n^{i t}; x)^{2} ⟶ \infty.

This follows by estimating the prime sum

\sum_{p \leq x} (1 + cos (t lo g p)) / p

.

Dirichlet convolution ( $*$ )

The Dirichlet convolution of arithmetic functions

f

and

g

is

(f * g) (n) = \sum_{d ∣ n} f (d) g (n / d) .

It is associative and commutative, its identity is the function supported at

1

, and the Möbius function is the convolution inverse of the constant-one function.

Dirichlet hyperbola method

The Dirichlet hyperbola method evaluates a summatory Dirichlet convolution by splitting the lattice points

ab \leq x

at

a, b \leq x

. For the divisor function,

\sum_{n \leq x} τ (n) = 2 \sum_{a \leq x} ⌊ \frac{x}{a} ⌋ - ⌊ x ⌋^{2} .

Divisor function ( $τ (n)$ )

The divisor function

τ (n) = \sum_{d ∣ n} 1

counts the positive integer divisors of

n

.

Von Mangoldt function ( $Λ (n)$ )

The Von Mangoldt function is

Λ (n) = lo g p

when

n

is a positive power of a prime number

p

, and is zero otherwise.

Von Mangoldt divisor identity

For every positive integer

n

,

\sum_{d ∣ n} Λ (d) = lo g n .

Equivalently, the constant-one function convolved with the Von Mangoldt function is the natural logarithm.

Möbius function

The Möbius function is

μ (1) = 1

, is zero on integers divisible by a prime square, and is

(- 1)^{r}

on products of

r

distinct primes.

Möbius inversion formula

If arithmetic functions

f, g

satisfy

g (n) = \sum_{d ∣ n} f (d),

then

f (n) = \sum_{d ∣ n} μ (d) g (n / d) .

Mertens function ( $M (x)$ )

The Mertens function is the summatory Möbius function

M (x) = \sum_{n \leq x} μ (n) .

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An arithmetic function is a mathematical function defined on the positive integers that takes real or complex values and often has significant implications in number theory. These functions can be classified into different categories based on their properties and applications. ### Key Characteristics: 1. **Domain**: The domain of an arithmetic function is usually the set of positive integers (denoted by \( \mathbb{Z}^+ \)).

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