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.

Additive functions

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

a complex vector space ,
an integrity ring ,
a commutative C - algebra .

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 .

literature

Jörg Brüdern : Introduction to analytical number theory. Springer-Verlag, 1995, ISBN 3-540-58821-3 .
Peter Bundschuh : Introduction to Number Theory. 5th edition. Springer-Verlag, 2002, ISBN 3-540-43579-4 .
Wolfgang Schwarz, Jürgen Spilker: Arithmetical Functions. Cambridge University Press, 1994, ISBN 0-521-42725-8 .
Paul McCarthy: Arithmetic Functions. Springer Spectrum, 2017, ISBN 978-3662537312 .

Web links

Arithmetic Function. In: Planetmath.org. (English).