In number theory, an arithmetic, arithmetical, or number-theoretic function [1] [2] is generally any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. [3] [4] [5] Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". [6] 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.
An arithmetic function a is
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
In this article, and mean that the sum or product is over all prime numbers:
and
Similarly, and mean that the sum or product is over all prime powers with strictly positive exponent (so k = 0 is not included):
The notations and mean that the sum or product is over all positive divisors of n, including 1 and n. For example, if n = 12, then
The notations can be combined: and mean that the sum or product is over all prime divisors of n. For example, if n = 18, then
and similarly and mean that the sum or product is over all prime powers dividing n. For example, if n = 24, then
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 < ... < pk are primes and the aj 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 νp(n) to be the exponent of the highest power of the prime p that divides n. That is, if p is one of the pi then νp(n) = ai, otherwise it is zero. Then
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 pi, ai, ω, and Ω.
σk(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).
Setting k = 0 in the second product gives
φ(n) , the Euler totient function, is the number of positive integers not greater than n that are coprime to n.
Jk(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) , 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) , 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", [7] (τ(n) is (2π)−12 times the nth Fourier coefficient in the q-expansion of the modular discriminant function) [8] it is included among the arithmetical functions because it is multiplicative and it occurs in identities involving certain σk(n) and rk(n) functions (because these are also coefficients in the expansion of modular forms).
cq(n) , Ramanujan's sum, is the sum of the nth powers of the primitive qth roots of unity:
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:
The Dedekind psi function, used in the theory of modular functions, is defined by the formula
λ(n) , the Liouville function, is defined by
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):
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,
ω(n), defined above as the number of distinct primes dividing n, is additive (see Prime omega function).
Ω(n) , defined above as the number of prime factors of n counted with multiplicities, is completely additive (see Prime omega function).
For a fixed prime p, νp(n), defined above as the exponent of the largest power of p dividing n, is completely additive.
, where is the arithmetic derivative.
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.
π(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.
A related function counts prime powers with weight 1 for primes, 1/2 for their squares, 1/3 for cubes, ... 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.
θ(x) and ψ(x), the Chebyshev functions, are defined as sums of the natural logarithms of the primes not exceeding x.
The Chebyshev function ψ(x) is the summation function of the von Mangoldt function just below.
Λ(n) , the von Mangoldt function, is 0 unless the argument n is a prime power pk, in which case it is the natural log of the prime p:
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:
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:
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.
rk(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.
Using the Heaviside notation for the derivative, the arithmetic derivative D(n) is a function such that
Given an arithmetic function a(n), its summation functionA(x) is defined by
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:
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 [9] is given by the divisor summatory function, the summation function of d(n), the number of divisors of n:
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
as x tends to infinity. The example above shows that d(n) has the average order log(n). [10]
Given an arithmetic function a(n), let Fa(s), for complex s, be the function defined by the corresponding Dirichlet series (where it converges): [11]
Fa(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:
Consider two arithmetic functions a and b and their respective generating functions Fa(s) and Fb(s). The product Fa(s)Fb(s) can be computed as follows:
It is a straightforward exercise to show that if c(n) is defined by
then
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:
Multiplying by the inverse of the zeta function gives the Möbius inversion formula:
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.
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:
For all (Lagrange's four-square theorem).
where the Kronecker symbol has the values
There is a formula for r3 in the section on class numbers below.
where ν = ν2(n). [21] [22] [23]
where [24]
Define the function σk*(n) as [25]
That is, if n is odd, σk*(n) is the sum of the kth powers of the divisors of n, that is, σk(n), 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.
Adopt the convention that Ramanujan's τ(x) = 0 if x is not an integer.
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 an and bn.
These formulas may be proved analytically (see Eisenstein series) or by elementary methods. [28]
Since σk(n) (for natural number k) and τ(n) are integers, the above formulas can be used to prove congruences [35] for the functions. See Ramanujan tau function for some examples.
Extend the domain of the partition function by setting p(0) = 1.
Peter Gustav Lejeune Dirichlet discovered formulas that relate the class number h of quadratic number fields to the Jacobi symbol. [37]
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). [38]
Extend the Jacobi symbol to accept even numbers in the "denominator" by defining the Kronecker symbol:
Then if D < −4 is a fundamental discriminant [39] [40]
There is also a formula relating r3 and h. Again, let D be a fundamental discriminant, D < −4. Then [41]
Let be the nth harmonic number. Then
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.
In 1965 P Kesava Menon proved [47]
This has been generalized by a number of mathematicians. For example,
In fact, if f is any arithmetical function [51] [52]
where stands for Dirichlet convolution.
Let m and n be distinct, odd, and positive. Then the Jacobi symbol satisfies the law of quadratic reciprocity:
Let D(n) be the arithmetic derivative. Then the logarithmic derivative
See Arithmetic derivative for details.
Let λ(n) be Liouville's function. Then
Let λ(n) be Carmichael's function. Then
See Multiplicative group of integers modulo n and Primitive root modulo n.
n | factorization | 𝜙(n) | ω(n) | Ω(n) | 𝜆(n) | 𝜇(n) | 𝛬(n) | π(n) | 𝜎0(n) | 𝜎1(n) | 𝜎2(n) | r2(n) | r3(n) | r4(n) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 4 | 6 | 8 |
2 | 2 | 1 | 1 | 1 | −1 | −1 | 0.69 | 1 | 2 | 3 | 5 | 4 | 12 | 24 |
3 | 3 | 2 | 1 | 1 | −1 | −1 | 1.10 | 2 | 2 | 4 | 10 | 0 | 8 | 32 |
4 | 22 | 2 | 1 | 2 | 1 | 0 | 0.69 | 2 | 3 | 7 | 21 | 4 | 6 | 24 |
5 | 5 | 4 | 1 | 1 | −1 | −1 | 1.61 | 3 | 2 | 6 | 26 | 8 | 24 | 48 |
6 | 2 · 3 | 2 | 2 | 2 | 1 | 1 | 0 | 3 | 4 | 12 | 50 | 0 | 24 | 96 |
7 | 7 | 6 | 1 | 1 | −1 | −1 | 1.95 | 4 | 2 | 8 | 50 | 0 | 0 | 64 |
8 | 23 | 4 | 1 | 3 | −1 | 0 | 0.69 | 4 | 4 | 15 | 85 | 4 | 12 | 24 |
9 | 32 | 6 | 1 | 2 | 1 | 0 | 1.10 | 4 | 3 | 13 | 91 | 4 | 30 | 104 |
10 | 2 · 5 | 4 | 2 | 2 | 1 | 1 | 0 | 4 | 4 | 18 | 130 | 8 | 24 | 144 |
11 | 11 | 10 | 1 | 1 | −1 | −1 | 2.40 | 5 | 2 | 12 | 122 | 0 | 24 | 96 |
12 | 22 · 3 | 4 | 2 | 3 | −1 | 0 | 0 | 5 | 6 | 28 | 210 | 0 | 8 | 96 |
13 | 13 | 12 | 1 | 1 | −1 | −1 | 2.56 | 6 | 2 | 14 | 170 | 8 | 24 | 112 |
14 | 2 · 7 | 6 | 2 | 2 | 1 | 1 | 0 | 6 | 4 | 24 | 250 | 0 | 48 | 192 |
15 | 3 · 5 | 8 | 2 | 2 | 1 | 1 | 0 | 6 | 4 | 24 | 260 | 0 | 0 | 192 |
16 | 24 | 8 | 1 | 4 | 1 | 0 | 0.69 | 6 | 5 | 31 | 341 | 4 | 6 | 24 |
17 | 17 | 16 | 1 | 1 | −1 | −1 | 2.83 | 7 | 2 | 18 | 290 | 8 | 48 | 144 |
18 | 2 · 32 | 6 | 2 | 3 | −1 | 0 | 0 | 7 | 6 | 39 | 455 | 4 | 36 | 312 |
19 | 19 | 18 | 1 | 1 | −1 | −1 | 2.94 | 8 | 2 | 20 | 362 | 0 | 24 | 160 |
20 | 22 · 5 | 8 | 2 | 3 | −1 | 0 | 0 | 8 | 6 | 42 | 546 | 8 | 24 | 144 |
21 | 3 · 7 | 12 | 2 | 2 | 1 | 1 | 0 | 8 | 4 | 32 | 500 | 0 | 48 | 256 |
22 | 2 · 11 | 10 | 2 | 2 | 1 | 1 | 0 | 8 | 4 | 36 | 610 | 0 | 24 | 288 |
23 | 23 | 22 | 1 | 1 | −1 | −1 | 3.14 | 9 | 2 | 24 | 530 | 0 | 0 | 192 |
24 | 23 · 3 | 8 | 2 | 4 | 1 | 0 | 0 | 9 | 8 | 60 | 850 | 0 | 24 | 96 |
25 | 52 | 20 | 1 | 2 | 1 | 0 | 1.61 | 9 | 3 | 31 | 651 | 12 | 30 | 248 |
26 | 2 · 13 | 12 | 2 | 2 | 1 | 1 | 0 | 9 | 4 | 42 | 850 | 8 | 72 | 336 |
27 | 33 | 18 | 1 | 3 | −1 | 0 | 1.10 | 9 | 4 | 40 | 820 | 0 | 32 | 320 |
28 | 22 · 7 | 12 | 2 | 3 | −1 | 0 | 0 | 9 | 6 | 56 | 1050 | 0 | 0 | 192 |
29 | 29 | 28 | 1 | 1 | −1 | −1 | 3.37 | 10 | 2 | 30 | 842 | 8 | 72 | 240 |
30 | 2 · 3 · 5 | 8 | 3 | 3 | −1 | −1 | 0 | 10 | 8 | 72 | 1300 | 0 | 48 | 576 |
31 | 31 | 30 | 1 | 1 | −1 | −1 | 3.43 | 11 | 2 | 32 | 962 | 0 | 0 | 256 |
32 | 25 | 16 | 1 | 5 | −1 | 0 | 0.69 | 11 | 6 | 63 | 1365 | 4 | 12 | 24 |
33 | 3 · 11 | 20 | 2 | 2 | 1 | 1 | 0 | 11 | 4 | 48 | 1220 | 0 | 48 | 384 |
34 | 2 · 17 | 16 | 2 | 2 | 1 | 1 | 0 | 11 | 4 | 54 | 1450 | 8 | 48 | 432 |
35 | 5 · 7 | 24 | 2 | 2 | 1 | 1 | 0 | 11 | 4 | 48 | 1300 | 0 | 48 | 384 |
36 | 22 · 32 | 12 | 2 | 4 | 1 | 0 | 0 | 11 | 9 | 91 | 1911 | 4 | 30 | 312 |
37 | 37 | 36 | 1 | 1 | −1 | −1 | 3.61 | 12 | 2 | 38 | 1370 | 8 | 24 | 304 |
38 | 2 · 19 | 18 | 2 | 2 | 1 | 1 | 0 | 12 | 4 | 60 | 1810 | 0 | 72 | 480 |
39 | 3 · 13 | 24 | 2 | 2 | 1 | 1 | 0 | 12 | 4 | 56 | 1700 | 0 | 0 | 448 |
40 | 23 · 5 | 16 | 2 | 4 | 1 | 0 | 0 | 12 | 8 | 90 | 2210 | 8 | 24 | 144 |
41 | 41 | 40 | 1 | 1 | −1 | −1 | 3.71 | 13 | 2 | 42 | 1682 | 8 | 96 | 336 |
42 | 2 · 3 · 7 | 12 | 3 | 3 | −1 | −1 | 0 | 13 | 8 | 96 | 2500 | 0 | 48 | 768 |
43 | 43 | 42 | 1 | 1 | −1 | −1 | 3.76 | 14 | 2 | 44 | 1850 | 0 | 24 | 352 |
44 | 22 · 11 | 20 | 2 | 3 | −1 | 0 | 0 | 14 | 6 | 84 | 2562 | 0 | 24 | 288 |
45 | 32 · 5 | 24 | 2 | 3 | −1 | 0 | 0 | 14 | 6 | 78 | 2366 | 8 | 72 | 624 |
46 | 2 · 23 | 22 | 2 | 2 | 1 | 1 | 0 | 14 | 4 | 72 | 2650 | 0 | 48 | 576 |
47 | 47 | 46 | 1 | 1 | −1 | −1 | 3.85 | 15 | 2 | 48 | 2210 | 0 | 0 | 384 |
48 | 24 · 3 | 16 | 2 | 5 | −1 | 0 | 0 | 15 | 10 | 124 | 3410 | 0 | 8 | 96 |
49 | 72 | 42 | 1 | 2 | 1 | 0 | 1.95 | 15 | 3 | 57 | 2451 | 4 | 54 | 456 |
50 | 2 · 52 | 20 | 2 | 3 | −1 | 0 | 0 | 15 | 6 | 93 | 3255 | 12 | 84 | 744 |
51 | 3 · 17 | 32 | 2 | 2 | 1 | 1 | 0 | 15 | 4 | 72 | 2900 | 0 | 48 | 576 |
52 | 22 · 13 | 24 | 2 | 3 | −1 | 0 | 0 | 15 | 6 | 98 | 3570 | 8 | 24 | 336 |
53 | 53 | 52 | 1 | 1 | −1 | −1 | 3.97 | 16 | 2 | 54 | 2810 | 8 | 72 | 432 |
54 | 2 · 33 | 18 | 2 | 4 | 1 | 0 | 0 | 16 | 8 | 120 | 4100 | 0 | 96 | 960 |
55 | 5 · 11 | 40 | 2 | 2 | 1 | 1 | 0 | 16 | 4 | 72 | 3172 | 0 | 0 | 576 |
56 | 23 · 7 | 24 | 2 | 4 | 1 | 0 | 0 | 16 | 8 | 120 | 4250 | 0 | 48 | 192 |
57 | 3 · 19 | 36 | 2 | 2 | 1 | 1 | 0 | 16 | 4 | 80 | 3620 | 0 | 48 | 640 |
58 | 2 · 29 | 28 | 2 | 2 | 1 | 1 | 0 | 16 | 4 | 90 | 4210 | 8 | 24 | 720 |
59 | 59 | 58 | 1 | 1 | −1 | −1 | 4.08 | 17 | 2 | 60 | 3482 | 0 | 72 | 480 |
60 | 22 · 3 · 5 | 16 | 3 | 4 | 1 | 0 | 0 | 17 | 12 | 168 | 5460 | 0 | 0 | 576 |
61 | 61 | 60 | 1 | 1 | −1 | −1 | 4.11 | 18 | 2 | 62 | 3722 | 8 | 72 | 496 |
62 | 2 · 31 | 30 | 2 | 2 | 1 | 1 | 0 | 18 | 4 | 96 | 4810 | 0 | 96 | 768 |
63 | 32 · 7 | 36 | 2 | 3 | −1 | 0 | 0 | 18 | 6 | 104 | 4550 | 0 | 0 | 832 |
64 | 26 | 32 | 1 | 6 | 1 | 0 | 0.69 | 18 | 7 | 127 | 5461 | 4 | 6 | 24 |
65 | 5 · 13 | 48 | 2 | 2 | 1 | 1 | 0 | 18 | 4 | 84 | 4420 | 16 | 96 | 672 |
66 | 2 · 3 · 11 | 20 | 3 | 3 | −1 | −1 | 0 | 18 | 8 | 144 | 6100 | 0 | 96 | 1152 |
67 | 67 | 66 | 1 | 1 | −1 | −1 | 4.20 | 19 | 2 | 68 | 4490 | 0 | 24 | 544 |
68 | 22 · 17 | 32 | 2 | 3 | −1 | 0 | 0 | 19 | 6 | 126 | 6090 | 8 | 48 | 432 |
69 | 3 · 23 | 44 | 2 | 2 | 1 | 1 | 0 | 19 | 4 | 96 | 5300 | 0 | 96 | 768 |
70 | 2 · 5 · 7 | 24 | 3 | 3 | −1 | −1 | 0 | 19 | 8 | 144 | 6500 | 0 | 48 | 1152 |
71 | 71 | 70 | 1 | 1 | −1 | −1 | 4.26 | 20 | 2 | 72 | 5042 | 0 | 0 | 576 |
72 | 23 · 32 | 24 | 2 | 5 | −1 | 0 | 0 | 20 | 12 | 195 | 7735 | 4 | 36 | 312 |
73 | 73 | 72 | 1 | 1 | −1 | −1 | 4.29 | 21 | 2 | 74 | 5330 | 8 | 48 | 592 |
74 | 2 · 37 | 36 | 2 | 2 | 1 | 1 | 0 | 21 | 4 | 114 | 6850 | 8 | 120 | 912 |
75 | 3 · 52 | 40 | 2 | 3 | −1 | 0 | 0 | 21 | 6 | 124 | 6510 | 0 | 56 | 992 |
76 | 22 · 19 | 36 | 2 | 3 | −1 | 0 | 0 | 21 | 6 | 140 | 7602 | 0 | 24 | 480 |
77 | 7 · 11 | 60 | 2 | 2 | 1 | 1 | 0 | 21 | 4 | 96 | 6100 | 0 | 96 | 768 |
78 | 2 · 3 · 13 | 24 | 3 | 3 | −1 | −1 | 0 | 21 | 8 | 168 | 8500 | 0 | 48 | 1344 |
79 | 79 | 78 | 1 | 1 | −1 | −1 | 4.37 | 22 | 2 | 80 | 6242 | 0 | 0 | 640 |
80 | 24 · 5 | 32 | 2 | 5 | −1 | 0 | 0 | 22 | 10 | 186 | 8866 | 8 | 24 | 144 |
81 | 34 | 54 | 1 | 4 | 1 | 0 | 1.10 | 22 | 5 | 121 | 7381 | 4 | 102 | 968 |
82 | 2 · 41 | 40 | 2 | 2 | 1 | 1 | 0 | 22 | 4 | 126 | 8410 | 8 | 48 | 1008 |
83 | 83 | 82 | 1 | 1 | −1 | −1 | 4.42 | 23 | 2 | 84 | 6890 | 0 | 72 | 672 |
84 | 22 · 3 · 7 | 24 | 3 | 4 | 1 | 0 | 0 | 23 | 12 | 224 | 10500 | 0 | 48 | 768 |
85 | 5 · 17 | 64 | 2 | 2 | 1 | 1 | 0 | 23 | 4 | 108 | 7540 | 16 | 48 | 864 |
86 | 2 · 43 | 42 | 2 | 2 | 1 | 1 | 0 | 23 | 4 | 132 | 9250 | 0 | 120 | 1056 |
87 | 3 · 29 | 56 | 2 | 2 | 1 | 1 | 0 | 23 | 4 | 120 | 8420 | 0 | 0 | 960 |
88 | 23 · 11 | 40 | 2 | 4 | 1 | 0 | 0 | 23 | 8 | 180 | 10370 | 0 | 24 | 288 |
89 | 89 | 88 | 1 | 1 | −1 | −1 | 4.49 | 24 | 2 | 90 | 7922 | 8 | 144 | 720 |
90 | 2 · 32 · 5 | 24 | 3 | 4 | 1 | 0 | 0 | 24 | 12 | 234 | 11830 | 8 | 120 | 1872 |
91 | 7 · 13 | 72 | 2 | 2 | 1 | 1 | 0 | 24 | 4 | 112 | 8500 | 0 | 48 | 896 |
92 | 22 · 23 | 44 | 2 | 3 | −1 | 0 | 0 | 24 | 6 | 168 | 11130 | 0 | 0 | 576 |
93 | 3 · 31 | 60 | 2 | 2 | 1 | 1 | 0 | 24 | 4 | 128 | 9620 | 0 | 48 | 1024 |
94 | 2 · 47 | 46 | 2 | 2 | 1 | 1 | 0 | 24 | 4 | 144 | 11050 | 0 | 96 | 1152 |
95 | 5 · 19 | 72 | 2 | 2 | 1 | 1 | 0 | 24 | 4 | 120 | 9412 | 0 | 0 | 960 |
96 | 25 · 3 | 32 | 2 | 6 | 1 | 0 | 0 | 24 | 12 | 252 | 13650 | 0 | 24 | 96 |
97 | 97 | 96 | 1 | 1 | −1 | −1 | 4.57 | 25 | 2 | 98 | 9410 | 8 | 48 | 784 |
98 | 2 · 72 | 42 | 2 | 3 | −1 | 0 | 0 | 25 | 6 | 171 | 12255 | 4 | 108 | 1368 |
99 | 32 · 11 | 60 | 2 | 3 | −1 | 0 | 0 | 25 | 6 | 156 | 11102 | 0 | 72 | 1248 |
100 | 22 · 52 | 40 | 2 | 4 | 1 | 0 | 0 | 25 | 9 | 217 | 13671 | 12 | 30 | 744 |
n | factorization | 𝜙(n) | ω(n) | Ω(n) | 𝜆(n) | 𝜇(n) | 𝛬(n) | π(n) | 𝜎0(n) | 𝜎1(n) | 𝜎2(n) | r2(n) | r3(n) | r4(n) |
In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the context of different conditions.
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The Liouville lambda function, denoted by λ(n) and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if n is the product of an even number of prime numbers, and −1 if it is the product of an odd number of primes.
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems proven by mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries of physical space.
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is the two-dimensional special case of Stokes' theorem.
In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process in which temperature is held constant. At constant temperature, the Helmholtz free energy is minimized at equilibrium.
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations.
In mathematics, a Dirichlet series is any series of the form
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.
In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell.
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
In mathematics, Fejér's theorem, named after Hungarian mathematician Lipót Fejér, states the following:
In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula
In number theory, an average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average".
In number theory, the prime omega functions and count the number of prime factors of a natural number Thereby counts each distinct prime factor, whereas the related function counts the total number of prime factors of honoring their multiplicity. That is, if we have a prime factorization of of the form for distinct primes , then the respective prime omega functions are given by and . These prime factor counting functions have many important number theoretic relations.
Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. The word calculus is a Latin word, meaning originally "small pebble"; as such pebbles were used for calculation, the meaning of the word has evolved and today usually means a method of computation. Meanwhile, calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the study of continuous change.