Arithmetic function explained

In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers.^{[1] [2] [3]} Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n".^[4] There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes.

An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n.

Arithmetic functions are often extremely irregular (see table), but some of them have series expansions in terms of Ramanujan's sum.

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(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(mn) = a(m)a(n) for all coprime natural numbers m and n.

Notation

In this article, $\backslash sum\_p\; f(p)$ and $\backslash prod\_p\; f(p)$ mean that the sum or product is over all prime numbers:$$\backslash sum\_p\; f(p)\; =\; f(2)\; +\; f(3)\; +\; f(5)\; +\; \backslash cdots$$and$$\backslash prod\_p\; f(p)=\; f(2)f(3)f(5)\backslash cdots.$$Similarly, $\backslash sum\_\; f(p^k)$ and $\backslash prod\_\; f(p^k)$ mean that the sum or product is over all prime powers with strictly positive exponent (so is not included):$$\backslash sum\_\; f(p^k)\; =\; \backslash sum\_p\backslash sum\_\; f(p^k)\; =\; f(2)\; +\; f(3)\; +\; f(4)\; +f(5)\; +f(7)+f(8)+f(9)+\backslash cdots.$$

The notations $\backslash sum\_\; f(d)$ and $\backslash prod\_\; f(d)$ mean that the sum or product is over all positive divisors of n, including 1 and n. For example, if, then$$\backslash prod\_\; f(d)\; =\; f(1)f(2)\; f(3)\; f(4)\; f(6)\; f(12).$$

The notations can be combined: $\backslash sum\_\; f(p)$ and $\backslash prod\_\; f(p)$ mean that the sum or product is over all prime divisors of n. For example, if n = 18, then$$\backslash sum\_\; f(p)\; =\; f(2)\; +\; f(3),$$and similarly $\backslash sum\_\; f(p^k)$ and $\backslash prod\_\; f(p^k)$ mean that the sum or product is over all prime powers dividing n. For example, if n = 24, then$$\backslash prod\_\; f(p^k)\; =\; f(2)\; f(3)\; f(4)\; f(8).$$

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

The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes:

where p₁ < p₂ < ... < p_k are primes and the a_j are positive integers. (1 is given by the empty product.)

It is often convenient to write this as an infinite product over all the primes, where all but a finite number have a zero exponent. Define the p-adic valuation ν_p(n) to be the exponent of the highest power of the prime p that divides n. That is, if p is one of the p_i then ν_p(n) = a_i, otherwise it is zero. Then$$n\; =\; \backslash prod\_p\; p^.$$

In terms of the above the prime omega functions ω and Ω are defined by

To avoid repetition, whenever possible formulas for the functions listed in this article are given in terms of n and the corresponding p_i, a_i, ω, and Ω.

Multiplicative functions

σ_k(n), τ(n), d(n) – divisor sums

σ_k(n) is the sum of the kth 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).

$$\backslash sigma\_k(n)\; =\; \backslash prod\_^\; \backslash frac=\; \backslash prod\_^\; \backslash left(1\; +\; p\_i^k\; +\; p\_i^\; +\; \backslash cdots\; +\; p\_i^\backslash right).$$

Setting k = 0 in the second product gives$$\backslash tau(n)\; =\; d(n)\; =\; (1\; +\; a\_)(1+a\_)\backslash cdots(1+a\_).$$

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.

$$\backslash varphi(n)\; =\; n\; \backslash prod\_\; \backslash left(1-\backslash frac\backslash right)=\; n\; \backslash left(\backslash frac\backslash right)\backslash left(\backslash frac\backslash right)\; \backslash cdots\; \backslash left(\backslash frac\backslash right).$$

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, .$$J\_k(n)\; =\; n^k\; \backslash prod\_\; \backslash left(1-\backslash frac\backslash right)=\; n^k\; \backslash left(\backslash frac\backslash right)\backslash left(\backslash frac\backslash right)\; \backslash cdots\; \backslash left(\backslash frac\backslash right).$$

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. (Because Ω(1) = ω(1) = 0.)

τ(n) – Ramanujan tau function

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

$$\backslash sum\_\backslash tau(n)q^n=q\backslash prod\_(1-q^n)^.$$

Although it is hard to say exactly what "arithmetical property of n" it "expresses",^[5] (τ(n) is (2π)⁻¹² times the nth Fourier coefficient in the q-expansion of the modular discriminant function)^[6] 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 (because these are also coefficients in the expansion of modular forms).

c_q(n) – Ramanujan's sum

c_q(n), Ramanujan's sum, is the sum of the nth powers of the primitive qth roots of unity:$$c\_q(n)\; =\; \backslash sum\_\; e^.$$

Even though it is defined as a sum of complex numbers (irrational for most values of q), it is an integer. For a fixed value of n it is multiplicative in q:

If q and r are coprime, then

ψ(n) - Dedekind psi function

The Dedekind psi function, used in the theory of modular functions, is defined by the formula$$\backslash psi(n)\; =\; n\; \backslash prod\_\backslash left(1+\backslash frac\backslash right).$$

Completely multiplicative functions

λ(n) – Liouville function

λ(n), the Liouville function, is defined by$$\backslash lambda\; (n)\; =\; (-1)^.$$

χ(n) – characters

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

The principal character (mod n) is denoted by χ₀(a) (or χ₁(a)). It is defined as

The quadratic character (mod n) is denoted by the Jacobi symbol for odd n (it is not defined for even n):$$\backslash left(\backslash frac\backslash right)\; =\; \backslash left(\backslash frac\backslash right)^\backslash left(\backslash frac\backslash right)^\backslash cdots\; \backslash left(\backslash frac\backslash right)^.$$

In this formula

is the Legendre symbol, defined for all integers a and all odd primes p by

Following the normal convention for the empty product,

Additive functions

ω(n) – distinct prime divisors

ω(n), defined above as the number of distinct primes dividing n, is additive (see Prime omega function).

Completely additive functions

Ω(n) – prime divisors

Ω(n), defined above as the number of prime factors of n counted with multiplicities, is completely additive (see Prime omega function).

ν_p(n) – p-adic valuation of an integer n

For a fixed prime p, ν_p(n), defined above as the exponent of the largest power of p dividing n, is completely additive.

Logarithmic derivative

}} \frac , where is the arithmetic derivative.

Neither multiplicative nor additive

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

These important functions (which are not arithmetic functions) are defined for non-negative real arguments, and are used in the various statements and proofs of the prime number theorem. They are summation functions (see the main section just below) of arithmetic functions which are neither multiplicative nor additive.

(x), the prime-counting function, is the number of primes not exceeding x. It is the summation function of the characteristic function of the prime numbers.$$\backslash pi(x)\; =\; \backslash sum\_\; 1$$

A related function counts prime powers with weight 1 for primes, 1/2 for their squares, 1/3 for cubes, etc. It is the summation function of the arithmetic function which takes the value 1/k on integers which are the k-th power of some prime number, and the value 0 on other integers.

$$\backslash Pi(x)\; =\; \backslash sum\_\backslash frac.$$

ϑ(x) and ψ(x), the Chebyshev functions, are defined as sums of the natural logarithms of the primes not exceeding x.$$\backslash vartheta(x)=\backslash sum\_\; \backslash log\; p,$$$$\backslash psi(x)\; =\; \backslash sum\_\; \backslash log\; p.$$

The second Chebyshev function ψ(x) is the summation function of the von Mangoldt function just below.

Λ(n) – von Mangoldt function

Λ(n), the von Mangoldt function, is 0 unless the argument n is a prime power, in which case it is the natural log of the prime p:

p(n) – partition function

p(n), the partition function, is the number of ways of representing n as a sum of positive integers, where two representations with the same summands in a different order are not counted as being different:$$p(n)\; =\; \backslash left|\backslash left\backslash \backslash right|.$$

λ(n) – Carmichael function

λ(n), the Carmichael function, is the smallest positive number such that

  for all a coprime to n. Equivalently, it is the least common multiple of the orders of the elements of the multiplicative group of integers modulo n.

For powers of odd primes and for 2 and 4, λ(n) is equal to the Euler totient function of n; for powers of 2 greater than 4 it is equal to one half of the Euler totient function of n:and for general n it is the least common multiple of λ of each of the prime power factors of n:$$\backslash lambda(p\_1^p\_2^\; \backslash dots\; p\_^)\; =\; \backslash operatorname[\backslash lambda(p\_1^\{a\_1\}),\backslash ;\backslash lambda(p\_2^\{a\_2\}),\backslash dots,\backslash lambda(p\_\{\backslash omega(n)\}^\{a\_\{\backslash omega(n)\}\})\; ].$$

h(n) – Class number

h(n), the class number function, is the order of the ideal class group of an algebraic extension of the rationals with discriminant n. The notation is ambiguous, as there are in general many extensions with the same discriminant. See quadratic field and cyclotomic field for classical examples.

r_k(n) – Sum of k squares

r_k(n) is the number of ways n can be represented as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the square roots are counted as different.

$$r\_k(n)\; =\; \backslash left|\backslash left\backslash \backslash right|$$

D(n) – Arithmetic derivative

Using the Heaviside notation for the derivative, the arithmetic derivative D(n) is a function such that

if n prime, and (the product rule)

Summation functions

Given an arithmetic function a(n), its summation function A(x) is defined by$$A(x)\; :=\; \backslash sum\_\; a(n)\; .$$A can be regarded as a function of a real variable. Given a positive integer m, A is constant along open intervals m < x < m + 1, and has a jump discontinuity at each integer for which a(m) ≠ 0.

Since such functions are often represented by series and integrals, to achieve pointwise convergence it is usual to define the value at the discontinuities as the average of the values to the left and right:$$A\_0(m)\; :=\; \backslash frac\; 1\; 2\; \backslash left(\backslash sum\_\; a(n)\; +\backslash sum\_\; a(n)\backslash right)\; =\; A(m)\; -\; \backslash frac\; 1\; 2\; a(m)\; .$$

Individual values of arithmetic functions may fluctuate wildly – as in most of the above examples. Summation functions "smooth out" these fluctuations. In some cases it may be possible to find asymptotic behaviour for the summation function for large x.

A classical example of this phenomenon^[7] is given by the divisor summatory function, the summation function of d(n), the number of divisors of n:$$\backslash liminf\_\; d(n)\; =\; 2$$$$\backslash limsup\_\backslash frac\; =\; \backslash log\; 2$$$$\backslash lim\_\backslash frac\; =\; 1.$$

An average order of an arithmetic function is some simpler or better-understood function which has the same summation function asymptotically, and hence takes the same values "on average". We say that g is an average order of f if$$\backslash sum\_\; f(n)\; \backslash sim\; \backslash sum\_\; g(n)$$

as x tends to infinity. The example above shows that d(n) has the average order log(n).^[8]

Dirichlet convolution

Given an arithmetic function a(n), let F_a(s), for complex s, be the function defined by the corresponding Dirichlet series (where it converges):^[9] $$F\_a(s)\; :=\; \backslash sum\_^\backslash infty\; \backslash frac\; .$$F_a(s) is called a generating function of a(n). The simplest such series, corresponding to the constant function a(n) = 1 for all n, is ζ(s) the Riemann zeta function.

The generating function of the Möbius function is the inverse of the zeta function:$$\backslash zeta(s)\backslash ,\backslash sum\_^\backslash infty\backslash frac=1,\; \backslash ;\backslash ;\backslash Re\; s\; >1.$$

Consider two arithmetic functions a and b and their respective generating functions F_a(s) and F_b(s). The product F_a(s)F_b(s) can be computed as follows:$$F\_a(s)F\_b(s)\; =\; \backslash left(\backslash sum\_^\backslash frac\; \backslash right)\backslash left(\backslash sum\_^\backslash frac\; \backslash right)\; .$$

It is a straightforward exercise to show that if c(n) is defined by$$c(n)\; :=\; \backslash sum\_\; a(i)b(j)\; =\; \backslash sum\_a(i)b\backslash left(\backslash frac\backslash right),$$then $$F\_c(s)\; =\; F\_a(s)\; F\_b(s).$$

This function c is called the Dirichlet convolution of a and b, and is denoted by

.

A particularly important case is convolution with the constant function a(n) = 1 for all n, corresponding to multiplying the generating function by the zeta function:$$g(n)\; =\; \backslash sum\_f(d).$$

Multiplying by the inverse of the zeta function gives the Möbius inversion formula:$$f(n)\; =\; \backslash sum\_\backslash mu\backslash left(\backslash frac\backslash right)g(d).$$

If f is multiplicative, then so is g. If f is completely multiplicative, then g is multiplicative, but may or may not be completely multiplicative.

Relations among the functions

There are a great many formulas connecting arithmetical functions with each other and with the functions of analysis, especially powers, roots, and the exponential and log functions. The page divisor sum identities contains many more generalized and related examples of identities involving arithmetic functions.

Here are a few examples:

Dirichlet convolutions

    where λ is the Liouville function.^{[10]      [11]}       Möbius inversion      ^[12]       Möbius inversion      ^{[13]      [14] [15]      [16]}       Möbius inversion             Möbius inversion             Möbius inversion           where λ is the Liouville function.      ^[17]       Möbius inversion

Sums of squares

For all

    (Lagrange's four-square theorem). ^[18] where the Kronecker symbol has the values

There is a formula for r₃ in the section on class numbers below.where .    ^{[19] [20] [21]}

$$r\_6(n)\; =\; 16\; \backslash sum\_\; \backslash chi\backslash left(\backslash frac\backslash right)d^2\; -\; 4\backslash sum\_\; \backslash chi(d)d^2,$$where

^[22]

Define the function as^[23]

That is, if n is odd, is the sum of the kth powers of the divisors of n, that is, and if n is even it is the sum of the kth powers of the even divisors of n minus the sum of the kth powers of the odd divisors of n.

   ^[24]

Adopt the convention that Ramanujan's if x is not an integer.

   ^[25]

Divisor sum convolutions

Here "convolution" does not mean "Dirichlet convolution" but instead refers to the formula for the coefficients of the product of two power series:

The sequence

is called the convolution or the Cauchy product of the sequences a_n and b_n.
These formulas may be proved analytically (see Eisenstein series) or by elementary methods.^{[26]    [27]    [28]    [29]    [30]}     where τ(n) is Ramanujan's function.    ^{[31] [32]}

Since σ_k(n) (for natural number k) and τ(n) are integers, the above formulas can be used to prove congruences^[33] for the functions. See Ramanujan tau function for some examples.

Extend the domain of the partition function by setting

   ^[34]   This recurrence can be used to compute p(n).

Class number related

Peter Gustav Lejeune Dirichlet discovered formulas that relate the class number h of quadratic number fields to the Jacobi symbol.^[35]

An integer D is called a fundamental discriminant if it is the discriminant of a quadratic number field. This is equivalent to D ≠ 1 and either a) D is squarefree and D ≡ 1 (mod 4) or b) D ≡ 0 (mod 4), D/4 is squarefree, and D/4 ≡ 2 or 3 (mod 4).^[36]

Extend the Jacobi symbol to accept even numbers in the "denominator" by defining the Kronecker symbol:

Then if D < −4 is a fundamental discriminant^{[37] [38]}

r\left(\frac\right)\\ & = \frac \sum_^

/2

\left(\frac\right).\end

There is also a formula relating r₃ and h. Again, let D be a fundamental discriminant, D < −4. Then^[39] $$r\_3(|D|)\; =\; 12\backslash left(1-\backslash left(\backslash frac\backslash right)\backslash right)h(D).$$

Prime-count related

Let

  be the nth harmonic number. Then   is true for every natural number n if and only if the Riemann hypothesis is true.    ^[40]

The Riemann hypothesis is also equivalent to the statement that, for all n > 5040, (where γ is the Euler–Mascheroni constant). This is Robin's theorem.

   ^{[41]    [42]    [43]    [44]}

Menon's identity

In 1965 P Kesava Menon proved^[45] $$\backslash sum\_\; \backslash gcd(k-1,n)=\backslash varphi(n)d(n).$$

This has been generalized by a number of mathematicians. For example,

• B. Sury^[46] $$$$

\sum_ \gcd(k_1-1,k_2,\dots,k_s,n)= \varphi(n)\sigma_(n).

• N. Rao^[47] $$$$

\sum_ \gcd(k_1-a_1,k_2-a_2,\dots,k_s-a_s,n)^s=J_s(n)d(n), where a₁, a₂, ..., a_s are integers, gcd(a₁, a₂, ..., a_s, n) = 1.

• László Fejes Tóth^[48] $$$$

\sum_ \gcd(k^2-1,m_1)\gcd(k^2-1,m_2)=\varphi(n)\sum_ \varphi(\gcd(d_1, d_2))2^, where m₁ and m₂ are odd, m = lcm(m₁, m₂).

In fact, if f is any arithmetical function^{[49] [50]} $$\backslash sum\_\; f(\backslash gcd(k-1,n))=\backslash varphi(n)\backslash sum\_\backslash frac,$$where

stands for Dirichlet convolution.

Miscellaneous

Let m and n be distinct, odd, and positive. Then the Jacobi symbol satisfies the law of quadratic reciprocity:$$\backslash left(\backslash frac\backslash right)\; \backslash left(\backslash frac\backslash right)\; =\; (-1)^.$$

Let D(n) be the arithmetic derivative. Then the logarithmic derivative $$\backslash frac\; =\; \backslash sum\_\; \backslash frac\; .$$ See Arithmetic derivative for details.

Let λ(n) be Liouville's function. Then

    and    

Let λ(n) be Carmichael's function. Then

    Further, See Multiplicative group of integers modulo n and Primitive root modulo n.     ^{[51] [52]    [53]    [54]}     Note that      ^{[55]    [56]}   Compare this with    ^{[57]    [58]}     where τ(n) is Ramanujan's function.    ^[59]

First 100 values of some arithmetic functions

n	factorization	(n)	ω(n)	Ω(n)	(n)	(n)	(n)	(n)	0(n)	1(n)	2(n)	r2(n)	r3(n)	r4(n)
1	style='text-align:center;'	1	1	0	0	1	1	0	0	1	1	1	4	6	8
2	style='text-align:center;'	2	1	1	1	−1	−1	0.69	1	2	3	5	4	12	24
3	style='text-align:center;'	3	2	1	1	−1	−1	1.10	2	2	4	10	0	8	32
4	style='text-align:center;'	22	2	1	2	1	0	0.69	2	3	7	21	4	6	24
5	style='text-align:center;'	5	4	1	1	−1	−1	1.61	3	2	6	26	8	24	48
6	style='text-align:center;'	2 · 3	2	2	2	1	1	0	3	4	12	50	0	24	96
7	style='text-align:center;'	7	6	1	1	−1	−1	1.95	4	2	8	50	0	0	64
8	style='text-align:center;'	23	4	1	3	−1	0	0.69	4	4	15	85	4	12	24
9	style='text-align:center;'	32	6	1	2	1	0	1.10	4	3	13	91	4	30	104
10	style='text-align:center;'	2 · 5	4	2	2	1	1	0	4	4	18	130	8	24	144
11	style='text-align:center;'	11	10	1	1	−1	−1	2.40	5	2	12	122	0	24	96
12	style='text-align:center;'	22 · 3	4	2	3	−1	0	0	5	6	28	210	0	8	96
13	style='text-align:center;'	13	12	1	1	−1	−1	2.56	6	2	14	170	8	24	112
14	style='text-align:center;'	2 · 7	6	2	2	1	1	0	6	4	24	250	0	48	192
15	style='text-align:center;'	3 · 5	8	2	2	1	1	0	6	4	24	260	0	0	192
16	style='text-align:center;'	24	8	1	4	1	0	0.69	6	5	31	341	4	6	24
17	style='text-align:center;'	17	16	1	1	−1	−1	2.83	7	2	18	290	8	48	144
18	style='text-align:center;'	2 · 32	6	2	3	−1	0	0	7	6	39	455	4	36	312
19	style='text-align:center;'	19	18	1	1	−1	−1	2.94	8	2	20	362	0	24	160
20	style='text-align:center;'	22 · 5	8	2	3	−1	0	0	8	6	42	546	8	24	144
21	style='text-align:center;'	3 · 7	12	2	2	1	1	0	8	4	32	500	0	48	256
22	style='text-align:center;'	2 · 11	10	2	2	1	1	0	8	4	36	610	0	24	288
23	style='text-align:center;'	23	22	1	1	−1	−1	3.14	9	2	24	530	0	0	192
24	style='text-align:center;'	23 · 3	8	2	4	1	0	0	9	8	60	850	0	24	96
25	style='text-align:center;'	52	20	1	2	1	0	1.61	9	3	31	651	12	30	248
26	style='text-align:center;'	2 · 13	12	2	2	1	1	0	9	4	42	850	8	72	336
27	style='text-align:center;'	33	18	1	3	−1	0	1.10	9	4	40	820	0	32	320
28	style='text-align:center;'	22 · 7	12	2	3	−1	0	0	9	6	56	1050	0	0	192
29	style='text-align:center;'	29	28	1	1	−1	−1	3.37	10	2	30	842	8	72	240
30	style='text-align:center;'	2 · 3 · 5	8	3	3	−1	−1	0	10	8	72	1300	0	48	576
31	style='text-align:center;'	31	30	1	1	−1	−1	3.43	11	2	32	962	0	0	256
32	style='text-align:center;'	25	16	1	5	−1	0	0.69	11	6	63	1365	4	12	24
33	style='text-align:center;'	3 · 11	20	2	2	1	1	0	11	4	48	1220	0	48	384
34	style='text-align:center;'	2 · 17	16	2	2	1	1	0	11	4	54	1450	8	48	432
35	style='text-align:center;'	5 · 7	24	2	2	1	1	0	11	4	48	1300	0	48	384
36	style='text-align:center;'	22 · 32	12	2	4	1	0	0	11	9	91	1911	4	30	312
37	style='text-align:center;'	37	36	1	1	−1	−1	3.61	12	2	38	1370	8	24	304
38	style='text-align:center;'	2 · 19	18	2	2	1	1	0	12	4	60	1810	0	72	480
39	style='text-align:center;'	3 · 13	24	2	2	1	1	0	12	4	56	1700	0	0	448
40	style='text-align:center;'	23 · 5	16	2	4	1	0	0	12	8	90	2210	8	24	144
41	style='text-align:center;'	41	40	1	1	−1	−1	3.71	13	2	42	1682	8	96	336
42	style='text-align:center;'	2 · 3 · 7	12	3	3	−1	−1	0	13	8	96	2500	0	48	768
43	style='text-align:center;'	43	42	1	1	−1	−1	3.76	14	2	44	1850	0	24	352
44	style='text-align:center;'	22 · 11	20	2	3	−1	0	0	14	6	84	2562	0	24	288
45	style='text-align:center;'	32 · 5	24	2	3	−1	0	0	14	6	78	2366	8	72	624
46	style='text-align:center;'	2 · 23	22	2	2	1	1	0	14	4	72	2650	0	48	576
47	style='text-align:center;'	47	46	1	1	−1	−1	3.85	15	2	48	2210	0	0	384
48	style='text-align:center;'	24 · 3	16	2	5	−1	0	0	15	10	124	3410	0	8	96
49	style='text-align:center;'	72	42	1	2	1	0	1.95	15	3	57	2451	4	54	456
50	style='text-align:center;'	2 · 52	20	2	3	−1	0	0	15	6	93	3255	12	84	744
51	style='text-align:center;'	3 · 17	32	2	2	1	1	0	15	4	72	2900	0	48	576
52	style='text-align:center;'	22 · 13	24	2	3	−1	0	0	15	6	98	3570	8	24	336
53	style='text-align:center;'	53	52	1	1	−1	−1	3.97	16	2	54	2810	8	72	432
54	style='text-align:center;'	2 · 33	18	2	4	1	0	0	16	8	120	4100	0	96	960
55	style='text-align:center;'	5 · 11	40	2	2	1	1	0	16	4	72	3172	0	0	576
56	style='text-align:center;'	23 · 7	24	2	4	1	0	0	16	8	120	4250	0	48	192
57	style='text-align:center;'	3 · 19	36	2	2	1	1	0	16	4	80	3620	0	48	640
58	style='text-align:center;'	2 · 29	28	2	2	1	1	0	16	4	90	4210	8	24	720
59	style='text-align:center;'	59	58	1	1	−1	−1	4.08	17	2	60	3482	0	72	480
60	style='text-align:center;'	22 · 3 · 5	16	3	4	1	0	0	17	12	168	5460	0	0	576
61	style='text-align:center;'	61	60	1	1	−1	−1	4.11	18	2	62	3722	8	72	496
62	style='text-align:center;'	2 · 31	30	2	2	1	1	0	18	4	96	4810	0	96	768
63	style='text-align:center;'	32 · 7	36	2	3	−1	0	0	18	6	104	4550	0	0	832
64	style='text-align:center;'	26	32	1	6	1	0	0.69	18	7	127	5461	4	6	24
65	style='text-align:center;'	5 · 13	48	2	2	1	1	0	18	4	84	4420	16	96	672
66	style='text-align:center;'	2 · 3 · 11	20	3	3	−1	−1	0	18	8	144	6100	0	96	1152
67	style='text-align:center;'	67	66	1	1	−1	−1	4.20	19	2	68	4490	0	24	544
68	style='text-align:center;'	22 · 17	32	2	3	−1	0	0	19	6	126	6090	8	48	432
69	style='text-align:center;'	3 · 23	44	2	2	1	1	0	19	4	96	5300	0	96	768
70	style='text-align:center;'	2 · 5 · 7	24	3	3	−1	−1	0	19	8	144	6500	0	48	1152
71	style='text-align:center;'	71	70	1	1	−1	−1	4.26	20	2	72	5042	0	0	576
72	style='text-align:center;'	23 · 32	24	2	5	−1	0	0	20	12	195	7735	4	36	312
73	style='text-align:center;'	73	72	1	1	−1	−1	4.29	21	2	74	5330	8	48	592
74	style='text-align:center;'	2 · 37	36	2	2	1	1	0	21	4	114	6850	8	120	912
75	style='text-align:center;'	3 · 52	40	2	3	−1	0	0	21	6	124	6510	0	56	992
76	style='text-align:center;'	22 · 19	36	2	3	−1	0	0	21	6	140	7602	0	24	480
77	style='text-align:center;'	7 · 11	60	2	2	1	1	0	21	4	96	6100	0	96	768
78	style='text-align:center;'	2 · 3 · 13	24	3	3	−1	−1	0	21	8	168	8500	0	48	1344
79	style='text-align:center;'	79	78	1	1	−1	−1	4.37	22	2	80	6242	0	0	640
80	style='text-align:center;'	24 · 5	32	2	5	−1	0	0	22	10	186	8866	8	24	144
81	style='text-align:center;'	34	54	1	4	1	0	1.10	22	5	121	7381	4	102	968
82	style='text-align:center;'	2 · 41	40	2	2	1	1	0	22	4	126	8410	8	48	1008
83	style='text-align:center;'	83	82	1	1	−1	−1	4.42	23	2	84	6890	0	72	672
84	style='text-align:center;'	22 · 3 · 7	24	3	4	1	0	0	23	12	224	10500	0	48	768
85	style='text-align:center;'	5 · 17	64	2	2	1	1	0	23	4	108	7540	16	48	864
86	style='text-align:center;'	2 · 43	42	2	2	1	1	0	23	4	132	9250	0	120	1056
87	style='text-align:center;'	3 · 29	56	2	2	1	1	0	23	4	120	8420	0	0	960
88	style='text-align:center;'	23 · 11	40	2	4	1	0	0	23	8	180	10370	0	24	288
89	style='text-align:center;'	89	88	1	1	−1	−1	4.49	24	2	90	7922	8	144	720
90	style='text-align:center;'	2 · 32 · 5	24	3	4	1	0	0	24	12	234	11830	8	120	1872
91	style='text-align:center;'	7 · 13	72	2	2	1	1	0	24	4	112	8500	0	48	896
92	style='text-align:center;'	22 · 23	44	2	3	−1	0	0	24	6	168	11130	0	0	576
93	style='text-align:center;'	3 · 31	60	2	2	1	1	0	24	4	128	9620	0	48	1024
94	style='text-align:center;'	2 · 47	46	2	2	1	1	0	24	4	144	11050	0	96	1152
95	style='text-align:center;'	5 · 19	72	2	2	1	1	0	24	4	120	9412	0	0	960
96	style='text-align:center;'	25 · 3	32	2	6	1	0	0	24	12	252	13650	0	24	96
97	style='text-align:center;'	97	96	1	1	−1	−1	4.57	25	2	98	9410	8	48	784
98	style='text-align:center;'	2 · 72	42	2	3	−1	0	0	25	6	171	12255	4	108	1368
99	style='text-align:center;'	32 · 11	60	2	3	−1	0	0	25	6	156	11102	0	72	1248
100	style='text-align:center;'	22 · 52	40	2	4	1	0	0	25	9	217	13671	12	30	744
n	factorization	(n)	ω(n)	Ω(n)	(n)	(n)	(n)	(n)	0(n)	1(n)	2(n)	r2(n)	r3(n)	r4(n)

References

• • • • • Book: Edwards , Harold . Harold Edwards (mathematician) . Fermat's Last Theorem . . New York . 1977 . 0-387-90230-9.

External links

• • Matthew Holden, Michael Orrison, Michael Varble Yet another Generalization of Euler's Totient Function
• Huard, Ou, Spearman, and Williams. Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions
• Dineva, Rosica, The Euler Totient, the Möbius, and the Divisor Functions
• László Tóth, Menon's Identity and arithmetical sums representing functions of several variables

Notes and References

1. Niven & Zuckerman, 4.2.
2. Nagell, I.9.
3. Bateman & Diamond, 2.1.
4. Hardy & Wright, intro. to Ch. XVI
5. Hardy, Ramanujan, § 10.2
6. Apostol, Modular Functions ..., § 1.15, Ch. 4, and ch. 6
7. Hardy & Wright, §§ 18.1–18.2
8. Book: Introduction to Analytic and Probabilistic Number Theory . Gérald Tenenbaum . Cambridge studies in advanced mathematics . 46 . . 36–55 . 1995 . 0-521-41261-7 .
9. Hardy & Wright, § 17.6, show how the theory of generating functions can be constructed in a purely formal manner with no attention paid to convergence.
10. Hardy & Wright, Thm. 263
11. Hardy & Wright, Thm. 63
12. see references at Jordan's totient function
13. Holden et al. in external links The formula is Gegenbauer's
14. Hardy & Wright, Thm. 288–290
15. Dineva in external links, prop. 4
16. Hardy & Wright, Thm. 264
17. Hardy & Wright, Thm. 296
18. Hardy & Wright, Thm. 278
19. Hardy & Wright, Thm. 386
20. Hardy, Ramanujan, eqs 9.1.2, 9.1.3
21. Koblitz, Ex. III.5.2
22. Hardy & Wright, § 20.13
23. Hardy, Ramanujan, § 9.7
24. Hardy, Ramanujan, § 9.13
25. Hardy, Ramanujan, § 9.17
26. Williams, ch. 13; Huard, et al. (external links).
27. Ramanujan, On Certain Arithmetical Functions, Table IV; Papers, p. 146
28. Koblitz, ex. III.2.8
29. Koblitz, ex. III.2.3
30. Koblitz, ex. III.2.2
31. Koblitz, ex. III.2.4
32. Apostol, Modular Functions ..., Ex. 6.10
33. Apostol, Modular Functions..., Ch. 6 Ex. 10
34. G.H. Hardy, S. Ramannujan, Asymptotic Formulæ in Combinatory Analysis, § 1.3; in Ramannujan, Papers p. 279
35. Landau, p. 168, credits Gauss as well as Dirichlet
36. Cohen, Def. 5.1.2
37. Cohen, Corr. 5.3.13
38. see Edwards, § 9.5 exercises for more complicated formulas.
39. Cohen, Prop 5.3.10
40. See Divisor function.
41. Hardy & Wright, eq. 22.1.2
42. See prime-counting functions.
43. Hardy & Wright, eq. 22.1.1
44. Hardy & Wright, eq. 22.1.3
45. László Tóth, Menon's Identity and Arithmetical Sums ..., eq. 1
46. Tóth, eq. 5
47. Tóth, eq. 3
48. Tóth, eq. 35
49. Tóth, eq. 2
50. Tóth states that Menon proved this for multiplicative f in 1965 and V. Sita Ramaiah for general f.
51. Hardy Ramanujan, eq. 3.10.3
52. Hardy & Wright, § 22.13
53. Hardy & Wright, Thm. 329
54. Hardy & Wright, Thms. 271, 272
55. Hardy & Wright, eq. 16.3.1
56. Ramanujan, Some Formulæ in the Analytic Theory of Numbers, eq. (C); Papers p. 133. A footnote says that Hardy told Ramanujan it also appears in an 1857 paper by Liouville.
57. Ramanujan, Some Formulæ in the Analytic Theory of Numbers, eq. (F); Papers p. 134
58. Apostol, Modular Functions ..., ch. 6 eq. 4
59. Apostol, Modular Functions ..., ch. 6 eq. 3