# Number theoretic function

A number theoretic or arithmetic function is a function that assigns a complex number to any positive natural number . These functions are used in number theory to describe and investigate properties of natural numbers, especially their divisibility .

## Special number theoretic functions

### Examples

The first values ​​of some number theoretic functions
n = φ (n) ω (n) Ω (n) λ (n) μ (n) Λ (n) π (n) σ 0 (n) σ 1 (n) σ 2 (n) r 2 (n) r 3 (n) r 4 (n)
1 1 1 0 0 1 1 0.00 0 1 1 1 4th 6th 8th
2 2 1 1 1 -1 -1 0.69 1 2 3 5 4th 12 24
3 3 2 1 1 -1 -1 1.10 2 2 4th 10 0 8th 32
4th 2 2 2 1 2 1 0 0.69 2 3 7th 21st 4th 6th 24
5 5 4th 1 1 -1 -1 1.61 3 2 6th 26th 8th 24 48
6th 2‧3 2 2 2 1 1 0.00 3 4th 12 50 0 24 96
7th 7th 6th 1 1 -1 -1 1.95 4th 2 8th 50 0 0 64
8th 2 3 4th 1 3 -1 0 0.69 4th 4th 15th 85 4th 12 24
9 3 2 6th 1 2 1 0 1.10 4th 3 13 91 4th 30th 104
10 2‧5 4th 2 2 1 1 0.00 4th 4th 18th 130 8th 24 144
11 11 10 1 1 -1 -1 2.40 5 2 12 122 0 24 96
12 2 2 ‧3 4th 2 3 -1 0 0.00 5 6th 28 210 0 8th 96
13 13 12 1 1 -1 -1 2.56 6th 2 14th 170 8th 24 112
14th 2‧7 6th 2 2 1 1 0.00 6th 4th 24 250 0 48 192
15th 3‧5 8th 2 2 1 1 0.00 6th 4th 24 260 0 0 192
16 2 4 8th 1 4th 1 0 0.69 6th 5 31 341 4th 6th 24
17th 17th 16 1 1 -1 -1 2.83 7th 2 18th 290 8th 48 144
18th 2‧3 2 6th 2 3 -1 0 0.00 7th 6th 39 455 4th 36 312
19th 19th 18th 1 1 -1 -1 2.94 8th 2 20th 362 0 24 160
20th 2 2 ‧5 8th 2 3 -1 0 0.00 8th 6th 42 546 8th 24 144
21st 3‧7 12 2 2 1 1 0.00 8th 4th 32 500 0 48 256
22nd 2‧11 10 2 2 1 1 0.00 8th 4th 36 610 0 24 288
23 23 22nd 1 1 -1 -1 3.14 9 2 24 530 0 0 192
24 2 3 ‧3 8th 2 4th 1 0 0.00 9 8th 60 850 0 24 96
25th 5 2 20th 1 2 1 0 1.61 9 3 31 651 12 30th 248
26th 2-13 12 2 2 1 1 0.00 9 4th 42 850 8th 72 336
27 3 3 18th 1 3 -1 0 1.10 9 4th 40 820 0 32 320
28 2 2 ‧7 12 2 3 -1 0 0.00 9 6th 56 1050 0 0 192
29 29 28 1 1 -1 -1 3.37 10 2 30th 842 8th 72 240
30th 2‧3‧5 8th 3 3 -1 -1 0.00 10 8th 72 1300 0 48 576
31 31 30th 1 1 -1 -1 3.43 11 2 32 962 0 0 256
32 2 5 16 1 5 -1 0 0.69 11 6th 63 1365 4th 12 24
33 3‧11 20th 2 2 1 1 0.00 11 4th 48 1220 0 48 384
34 2-17 16 2 2 1 1 0.00 11 4th 54 1450 8th 48 432
35 5-7 24 2 2 1 1 0.00 11 4th 48 1300 0 48 384
36 2 2 ‧3 2 12 2 4th 1 0 0.00 11 9 91 1911 4th 30th 312
37 37 36 1 1 -1 -1 3.61 12 2 38 1370 8th 24 304
38 2-19 18th 2 2 1 1 0.00 12 4th 60 1810 0 72 480
39 3-13 24 2 2 1 1 0.00 12 4th 56 1700 0 0 448
40 2 3 ‧5 16 2 4th 1 0 0.00 12 8th 90 2210 8th 24 144

Important arithmetic functions are

• the identical function and their powers${\ displaystyle I (n): = n}$${\ displaystyle I ^ {r} (n) = n ^ {r},}$
• the Dirichlet characters ${\ displaystyle \ chi _ {k} (n),}$
• the divider functions
${\ displaystyle \ qquad \ sigma _ {k} (n): = \ sum _ {d | n} d ^ {k},}$special ,${\ displaystyle \ sigma (n): = \ sigma _ {1} (n) = \ sum _ {d | n} d}$
which indicate the sum of all divisors or the -th powers of all divisors of a number and${\ displaystyle k}$${\ displaystyle n}$
• the number of divisors function that indicates how many divisors the number has,${\ displaystyle d (n): = \ sigma _ {0} (n)}$${\ displaystyle n}$
• the Euler's φ function that the number of indicating natural prime numbers that are not greater than are${\ displaystyle n}$${\ displaystyle n}$
• the Liouville function ,${\ displaystyle \ lambda (n)}$
• the order , i.e. the number of (not necessarily different) prime factors of , as well as the number of different prime factors,${\ displaystyle \ Omega (n)}$${\ displaystyle n}$${\ displaystyle \ omega (n)}$
• the Dedekind psi function ,
• the Möbius μ-function (see the paragraph on convolution below),
• the isomorphism type number function ,${\ displaystyle a (n)}$
• the p-adic exponent weighting ${\ displaystyle \ nu _ {p} (n),}$
• the prime number function that specifies the number of prime numbers that are not greater than ,${\ displaystyle \ pi (n),}$${\ displaystyle n}$
• the Smarandache function ,
• the Chebyshev function ,
• the Mangoldt function ,${\ displaystyle \ Lambda (n)}$
• the sum of squares functions as the number of representations of a given natural number as the sum of squares of integers.${\ displaystyle r_ {k} (n)}$${\ displaystyle n}$${\ displaystyle k}$

### Multiplicative functions

A number theoretic function is called multiplicative, if for coprime numbers and always holds and does not vanish, which is equivalent to . It is called completely multiplicative, also strictly or strictly multiplicative, if this also applies to non-prime numbers. So every fully multiplicative function is multiplicative. A multiplicative function can be represented as ${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle f (ab) = f (a) \ cdot f (b)}$${\ displaystyle f (1)}$${\ displaystyle f (1) = 1}$

${\ displaystyle f (n) = \ prod _ {p \ in \ mathbb {P}} f \ left (p ^ {\ nu _ {p} (n)} \ right),}$

d. H. a multiplicative function is completely determined by the values ​​it takes for prime powers.

• Of the functions listed above as examples, the identity and its powers as well as the Dirichlet characters are completely multiplicative, the number function, the divisor functions and Euler's φ function are multiplicative. The prime number function and the exponent weighting are not multiplicative.
• The (pointwise) product of two (completely) multiplicative functions is again (completely) multiplicative.

A number-theoretic function is called additive if and always holds for coprime numbers . It is called completely additive, also strictly or strictly additive, if this also applies to non-prime numbers. ${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle f (ab) = f (a) + f (b)}$

An example of an additive function is the -adic exponent weighting. Any multiplicative function that does not vanish anywhere can be constructed into an additive function by taking the result logarithmically. More precisely: If is (completely) multiplicative and always , then is a (completely) additive function. Occasionally, a (complex) logarithm of a number-theoretic function that does not vanish anywhere (without amount) is formed. However, caution should be exercised because of the various branches of the complex logarithm . ${\ displaystyle p}$${\ displaystyle f}$${\ displaystyle f (n) \ neq 0}$${\ displaystyle \ log (| f |)}$${\ displaystyle \ operatorname {Log} (f)}$

## folding

The convolution of number theoretic functions is also called Dirichlet convolution according to Dirichlet . For other meanings of the word in mathematics, see the article Convolution (mathematics) .

### definition

The Dirichlet convolution of two number theoretic functions is defined by

${\ displaystyle (f * g) (n): = \ sum _ {d \ mid n} f \! \ left ({\ frac {n} {d}} \ right) g (d), \ quad n \ in \ mathbb {N},}$

where the sum extends over all (real and improper, positive) divisors of . ${\ displaystyle n}$

The summation function of a number theoretic function is defined by , where the constant function is denoted by the function value  , thus ${\ displaystyle f}$${\ displaystyle F: = f * I ^ {0}}$${\ displaystyle I ^ {0}}$${\ displaystyle 1}$

${\ displaystyle F (n) = (f * I ^ {0}) (n) = \ sum _ {d \ mid n} f (d), \ quad n \ in \ mathbb {N}.}$

One can show that the convolution operation is invertible; its inverse is the (multiplicative) Möbius function . This leads to Möbius' inverse formula , with which one can recover a number-theoretic function from its summation function. ${\ displaystyle I ^ {0}}$ ${\ displaystyle \ mu}$

### Properties of the fold

• The convolution of two multiplicative functions is multiplicative.
• The convolution of two fully multiplicative functions need not be fully multiplicative.
• Every number theoretic function that does not vanish at that point  has an inverse with respect to the convolution operation.${\ displaystyle f}$${\ displaystyle 1}$
• This convolution inverse is multiplicative if and only if is multiplicative.${\ displaystyle f}$
• The convolution inverse of a fully multiplicative function is multiplicative, but generally not fully multiplicative.
• The neutral element of the convolution operation is the function defined by and for all${\ displaystyle \ eta (1) = 1}$${\ displaystyle \ eta (n) = 0}$${\ displaystyle n> 1}$${\ displaystyle \ eta.}$

### Algebraic structure

• The set of number theoretic functions forms with component-wise addition, scalar multiplication and convolution as internal multiplication
• The multiplicative group of this ring consists of the number theoretic functions that do not disappear at this point  .${\ displaystyle 1}$
• The set of multiplicative functions is a real subset of this group.

### Separation from the space of complex number sequences

With the complex scalar multiplication , the component-wise addition and - instead of the convolution - the component-wise multiplication, the set of number-theoretic functions also forms a commutative C -algebra, the algebra of the formal (not necessarily convergent) complex number sequences. However, this canonical structure as a mapping space is of little interest in number theory.

As a complex vector space (i.e. without internal multiplication), this sequence space is identical to the space of number theoretic functions.

## Connection with Dirichlet series

A formal Dirichlet series can be assigned to each number theoretic function. The convolution then becomes the multiplication of rows. This construction is described in more detail in the article on Dirichlet series .