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Arithmetic function

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. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". 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, $\sum_p f(p)$ and $\prod_p f(p)$  mean that the sum or product is over all prime numbers: $$\sum_p f(p) = f(2) + f(3) + f(5) + \cdots$$ and $$\prod_p f(p)= f(2)f(3)f(5)\cdots.$$ Similarly, $\sum_{p^k} f(p^k)$ and $\prod_{p^k} f(p^k)$  mean that the sum or product is over all prime powers with strictly positive exponent (so $k = 0$ is not included): $$\sum_{p^k} f(p^k) = \sum_p\sum_{k > 0} f(p^k) = f(2) + f(3) + f(4) +f(5) +f(7)+f(8)+f(9)+\cdots.$$

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

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

Ω(n), ω(n), νp(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: $$ n = p_1^{a_1}\cdots p_k^{a_k} $$ 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 $$n = \prod_p p^{\nu_p(n)}.$$

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), τ(n), d(n) – divisor sums
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).

$$\sigma_k(n) = \prod_{i=1}^{\omega(n)} \frac{p_i^{(a_i+1)k}-1}{p_i^k-1}= \prod_{i=1}^{\omega(n)} \left(1 + p_i^k + p_i^{2k} + \cdots + p_i^{a_i k}\right).$$

Setting k = 0 in the second product gives $$\tau(n) = d(n) = (1 + a_{1})(1+a_{2})\cdots(1+a_{\omega(n)}).$$

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.

$$\varphi(n) = n \prod_{p\mid n} \left(1-\frac{1}{p}\right) = n \left(\frac{p_1 - 1}{p_1}\right)\left(\frac{p_2 - 1}{p_2}\right) \cdots \left(\frac{p_{\omega(n)} - 1}{p_{\omega(n)}}\right) .$$

Jk(n) – Jordan totient function
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, $φ(n) = J_{1}(n)$. $$J_k(n) = n^k \prod_{p\mid n} \left(1-\frac{1}{p^k}\right) = n^k \left(\frac{p^k_1 - 1}{p^k_1}\right)\left(\frac{p^k_2 - 1}{p^k_2}\right) \cdots \left(\frac{p^k_{\omega(n)} - 1}{p^k_{\omega(n)}}\right) .$$

μ(n) – Möbius function
μ(n), the Möbius function, is important because of the Möbius inversion formula. See Dirichlet convolution, below.

$$\mu(n)=\begin{cases} (-1)^{\omega(n)}=(-1)^{\Omega(n)} &\text{if }\; \omega(n) = \Omega(n)\\ 0&\text{if }\;\omega(n) \ne \Omega(n). \end{cases}$$

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:

$$\sum_{n\geq 1}\tau(n)q^n=q\prod_{n\geq 1}(1-q^n)^{24}.$$

Although it is hard to say exactly what "arithmetical property of n" it "expresses", (τ(n) is (2π)−12 times the nth Fourier coefficient in the q-expansion of the modular discriminant function) 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
cq(n), Ramanujan's sum, is the sum of the nth powers of the primitive qth roots of unity: $$c_q(n) = \sum_{\stackrel{1\le a\le q}{ \gcd(a,q)=1}} e^{2 \pi i \tfrac{a}{q} n}.$$

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 $$c_q(n)c_r(n)=c_{qr}(n).$$

ψ(n) - Dedekind psi function
The Dedekind psi function, used in the theory of modular functions, is defined by the formula $$ \psi(n) = n \prod_{p|n}\left(1+\frac{1}{p}\right).$$

λ(n) – Liouville function
λ(n), the Liouville function, is defined by $$\lambda (n) = (-1)^{\Omega(n)}.$$

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 $$ \chi_0(a) = \begin{cases} 1 & \text{if } \gcd(a,n) = 1, \\ 0 & \text{if } \gcd(a,n) \ne 1. \end{cases} $$

The quadratic character (mod n) is denoted by the Jacobi symbol for odd n (it is not defined for even n): $$\left(\frac{a}{n}\right) = \left(\frac{a}{p_1}\right)^{a_1}\left(\frac{a}{p_2}\right)^{a_2}\cdots \left(\frac{a}{p_{\omega(n)}}\right)^{a_{\omega(n)}}.$$

In this formula $$(\tfrac{a}{p})$$ is the Legendre symbol, defined for all integers a and all odd primes p by $$ \left(\frac{a}{p}\right) = \begin{cases} \;\;\,0 & \text{if } a \equiv 0 \pmod p, \\+1 & \text{if }a \not\equiv 0\pmod p \text{ and for some integer }x, \;a\equiv x^2\pmod p \\-1 & \text{if there is no such } x. \end{cases}$$

Following the normal convention for the empty product, $$\left(\frac{a}{1}\right) = 1.$$

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

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).

νp(n) – p-adic valuation of an integer n
For a fixed prime p, νp(n), defined above as the exponent of the largest power of p dividing n, is completely additive.

Logarithmic derivative
$$\operatorname{ld}(n)=\frac{D(n)}{n} = \sum_{\stackrel{p\mid n}{p\text{ prime}}} \frac {v_p(n)} {p}$$, where $$D(n)$$ is the arithmetic derivative.

$\pi$(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.

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. $$\pi(x) = \sum_{p \le x} 1$$

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.

$$\Pi(x) = \sum_{p^k\le x}\frac{1}{k}.$$

θ(x) and ψ(x), the Chebyshev functions, are defined as sums of the natural logarithms of the primes not exceeding x. $$\vartheta(x)=\sum_{p\le x} \log p,$$ $$ \psi(x) = \sum_{p^k\le x} \log p.$$

The 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 $p^{k}$, in which case it is the natural log of the prime p: $$\Lambda(n) = \begin{cases} \log p &\text{if } n = 2,3,4,5,7,8,9,11,13,16,\ldots=p^k \text{ is a prime power}\\ 0&\text{if } n=1,6,10,12,14,15,18,20,21,\dots \;\;\;\;\text{ is not a prime power}. \end{cases}$$

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: $$p(n) = \left|\left\{ (a_1, a_2,\dots a_k): 0 < a_1 \le a_2 \le \cdots \le a_k\; \land \;n=a_1+a_2+\cdots +a_k \right\}\right|.$$

λ(n) – Carmichael function
λ(n), the Carmichael function, is the smallest positive number such that $$a^{\lambda(n)}\equiv 1 \pmod{n}$$  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: $$\lambda(n) = \begin{cases} \;\;\phi(n) &\text{if }n = 2,3,4,5,7,9,11,13,17,19,23,25,27,\dots\\ \tfrac 1 2 \phi(n)&\text{if }n=8,16,32,64,\dots \end{cases}$$ and for general n it is the least common multiple of λ of each of the prime power factors of n: $$\lambda(p_1^{a_1}p_2^{a_2} \dots p_{\omega(n)}^{a_{\omega(n)}}) = \operatorname{lcm}[\lambda(p_1^{a_1}),\;\lambda(p_2^{a_2}),\dots,\lambda(p_{\omega(n)}^{a_{\omega(n)}}) ].$$

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.

rk(n) – Sum of k squares
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.

$$r_k(n) = \left|\left\{(a_1, a_2,\dots,a_k):n=a_1^2+a_2^2+\cdots+a_k^2\right\}\right|$$

D(n) – Arithmetic derivative
Using the Heaviside notation for the derivative, the arithmetic derivative D(n) is a function such that
 * $$ D(n) = 1$$ if n prime, and
 * $$D(mn) = m D(n) + D(m) n$$ (the product rule)

Summation functions
Given an arithmetic function a(n), its summation function A(x) is defined by $$ A(x) := \sum_{n \le x} a(n) .$$ 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: $$ A_0(m) := \frac 1 2 \left(\sum_{n < m} a(n) +\sum_{n \le m} a(n)\right) = A(m) - \frac 1 2 a(m) .$$

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 is given by the divisor summatory function, the summation function of d(n), the number of divisors of n: $$\liminf_{n\to\infty} d(n) = 2$$ $$\limsup_{n\to\infty}\frac{\log d(n) \log\log n}{\log n} = \log 2$$ $$\lim_{n\to\infty}\frac{d(1) + d(2)+ \cdots +d(n)}{\log(1) + \log(2)+ \cdots +\log(n)} = 1.$$

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 $$ \sum_{n \le x} f(n) \sim \sum_{n \le x} g(n) $$

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

Dirichlet convolution
Given an arithmetic function a(n), let Fa(s), for complex s, be the function defined by the corresponding Dirichlet series (where it converges): $$ F_a(s) := \sum_{n=1}^\infty \frac{a(n)}{n^s} .$$ 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: $$\zeta(s)\,\sum_{n=1}^\infty\frac{\mu(n)}{n^s}=1, \;\;\Re s >1.$$

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: $$ F_a(s)F_b(s) = \left( \sum_{m=1}^{\infty}\frac{a(m)}{m^s} \right)\left( \sum_{n=1}^{\infty}\frac{b(n)}{n^s} \right). $$

It is a straightforward exercise to show that if c(n) is defined by $$ c(n) := \sum_{ij = n} a(i)b(j) = \sum_{i\mid n}a(i)b\left(\frac{n}{i}\right), $$ 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 $$a*b$$.

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: $$g(n) = \sum_{d \mid n}f(d).$$

Multiplying by the inverse of the zeta function gives the Möbius inversion formula: $$f(n) = \sum_{d\mid n}\mu\left(\frac{n}{d}\right)g(d).$$

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

\sum_{\delta\mid n}\mu(\delta)= \sum_{\delta\mid n}\lambda\left(\frac{n}{\delta}\right)|\mu(\delta)|= \begin{cases} 1 & \text{if } n=1\\ 0 & \text{if } n\ne1 \end{cases} $$    where λ is the Liouville function.

 * $$\sum_{\delta\mid n}\varphi(\delta) = n.$$
 * $$\varphi(n)

=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)\delta =n\sum_{\delta\mid n}\frac{\mu(\delta)}{\delta}. $$       Möbius inversion
 * $$\sum_{d \mid n } J_k(d) = n^k.$$

J_k(n) =\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)\delta^k =n^k\sum_{\delta\mid n}\frac{\mu(\delta)}{\delta^k}. $$       Möbius inversion
 * $$\sum_{\delta\mid n}\delta^sJ_r(\delta)J_s\left(\frac{n}{\delta}\right) = J_{r+s}(n)$$
 * $$\sum_{\delta\mid n}\varphi(\delta)d\left(\frac{n}{\delta}\right) = \sigma(n).$$
 * $$\sum_{\delta\mid n}|\mu(\delta)| = 2^{\omega(n)}.$$
 * $$|\mu(n)|=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)2^{\omega(\delta)}.$$       Möbius inversion
 * $$\sum_{\delta\mid n}2^{\omega(\delta)}=d(n^2).$$
 * $$2^{\omega(n)}=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)d(\delta^2).$$       Möbius inversion
 * $$\sum_{\delta\mid n}d(\delta^2)=d^2(n).$$
 * $$d(n^2)=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)d^2(\delta).$$       Möbius inversion
 * $$\sum_{\delta\mid n}d\left(\frac{n}{\delta}\right)2^{\omega(\delta)}=d^2(n).$$
 * $$\sum_{\delta\mid n}\lambda(\delta)=\begin{cases}

&1\text{ if } n \text{ is a square }\\ &0\text{ if } n \text{ is not square.} \end{cases}$$      where λ is the Liouville function.
 * $$\sum_{\delta\mid n}\Lambda(\delta) = \log n.$$
 * $$\Lambda(n)=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)\log(\delta).$$       Möbius inversion

Sums of squares
For all $$k \ge 4,\;\;\; r_k(n) > 0.$$    (Lagrange's four-square theorem).

r_2(n) = 4\sum_{d\mid n}\left(\frac{-4}{d}\right), $$ where the Kronecker symbol has the values

\left(\frac{-4}{n}\right) = \begin{cases} +1&\text{if }n\equiv 1 \pmod 4 \\ -1&\text{if }n\equiv 3 \pmod 4\\ \;\;\;0&\text{if }n\text{ is even}.\\ \end{cases} $$

There is a formula for r3 in the section on class numbers below. $$ r_4(n) = 8 \sum_{\stackrel{d\mid n}{ 4\, \nmid \,d}}d = 8 (2+(-1)^n)\sum_{\stackrel{d\mid n}{ 2\, \nmid \,d}}d = \begin{cases} 8\sigma(n)&\text{if } n \text{ is odd }\\ 24\sigma\left(\frac{n}{2^\nu}\right)&\text{if } n \text{ is even } \end{cases}, $$ where $ν = ν_{2}(n)$.

$$r_6(n) = 16 \sum_{d\mid n} \chi\left(\frac{n}{d}\right)d^2 - 4\sum_{d\mid n} \chi(d)d^2,$$ where $$ \chi(n) = \left(\frac{-4}{n}\right).$$

Define the function $σ_{k}^{*}(n)$ as $$\sigma_k^*(n) = (-1)^{n}\sum_{d\mid n}(-1)^d d^k= \begin{cases} \sum_{d\mid n} d^k=\sigma_k(n)&\text{if } n \text{ is odd }\\ \sum_{\stackrel{d\mid n}{ 2\, \mid \,d}}d^k -\sum_{\stackrel{d\mid n}{ 2\, \nmid \,d}}d^k&\text{if } n \text{ is even}. \end{cases} $$

That is, if n is odd, $σ_{k}^{*}(n)$ is the sum of the kth powers of the divisors of n, that is, $σ_{k}(n),$ 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.

 * $$r_8(n) = 16\sigma_3^*(n).$$

Adopt the convention that Ramanujan's $τ(x) = 0$ if x is not an integer.

r_{24}(n) = \frac{16}{691}\sigma_{11}^*(n) + \frac{128}{691}\left\{ (-1)^{n-1}259\tau(n)-512\tau\left(\frac{n}{2}\right)\right\} $$

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:

 * $$ \left(\sum_{n=0}^\infty a_n x^n\right)\left(\sum_{n=0}^\infty b_n x^n\right)

= \sum_{i=0}^\infty \sum_{j=0}^\infty a_i b_j x^{i+j} = \sum_{n=0}^\infty \left(\sum_{i=0}^n a_i b_{n-i}\right) x^n = \sum_{n=0}^\infty c_n x^n .$$

The sequence $$c_n = \sum_{i=0}^n a_i b_{n-i}$$ 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.

\sigma_3(n) = \frac{1}{5}\left\{6n\sigma_1(n)-\sigma_1(n) + 12\sum_{0<k<n}\sigma_1(k)\sigma_1(n-k)\right\}. $$

\sigma_5(n) = \frac{1}{21}\left\{10(3n-1)\sigma_3(n)+\sigma_1(n) + 240\sum_{0<k<n}\sigma_1(k)\sigma_3(n-k)\right\}. $$

\begin{align} \sigma_7(n) &=\frac{1}{20}\left\{21(2n-1)\sigma_5(n)-\sigma_1(n) + 504\sum_{0<k<n}\sigma_1(k)\sigma_5(n-k)\right\}\\ &=\sigma_3(n) + 120\sum_{0<k<n}\sigma_3(k)\sigma_3(n-k). \end{align} $$

\begin{align} \sigma_9(n) &= \frac{1}{11}\left\{10(3n-2)\sigma_7(n)+\sigma_1(n) + 480\sum_{0<k<n}\sigma_1(k)\sigma_7(n-k)\right\}\\ &= \frac{1}{11}\left\{21\sigma_5(n)-10\sigma_3(n) + 5040\sum_{0<k<n}\sigma_3(k)\sigma_5(n-k)\right\}. \end{align} $$

\tau(n) = \frac{65}{756}\sigma_{11}(n) + \frac{691}{756}\sigma_{5}(n) - \frac{691}{3}\sum_{0<k<n}\sigma_5(k)\sigma_5(n-k), $$    where τ(n) is Ramanujan's function.

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

Extend the domain of the partition function by setting $p(0) = 1.$

p(n)=\frac{1}{n}\sum_{1\le k\le n}\sigma(k)p(n-k). $$     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.

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).

Extend the Jacobi symbol to accept even numbers in the "denominator" by defining the Kronecker symbol: $$\left(\frac{a}{2}\right) = \begin{cases} \;\;\,0&\text{ if } a \text{ is even} \\(-1)^{\frac{a^2-1}{8}}&\text{ if }a \text{ is odd. } \end{cases}$$

Then if D < −4 is a fundamental discriminant $$\begin{align} h(D) & = \frac{1}{D} \sum_{r=1}^{|D|}r\left(\frac{D}{r}\right)\\ & = \frac{1}{2-\left(\tfrac{D}{2}\right)} \sum_{r=1}^{|D|/2}\left(\frac{D}{r}\right). \end{align}$$

There is also a formula relating r3 and h. Again, let D be a fundamental discriminant, D < −4. Then $$r_3(|D|) = 12\left(1-\left(\frac{D}{2}\right)\right)h(D).$$

Prime-count related
Let $$H_n = 1 + \frac 1 2 + \frac 1 3 + \cdots +\frac{1}{n}$$  be the nth harmonic number. Then

 * $$ \sigma(n) \le H_n + e^{H_n}\log H_n$$  is true for every natural number n if and only if the Riemann hypothesis is true.

The Riemann hypothesis is also equivalent to the statement that, for all n > 5040, $$\sigma(n) < e^\gamma n \log \log n $$ (where γ is the Euler–Mascheroni constant). This is Robin's theorem.

 * $$\sum_{p}\nu_p(n) = \Omega(n).$$
 * $$\psi(x)=\sum_{n\le x}\Lambda(n).$$
 * $$\Pi(x)= \sum_{n\le x}\frac{\Lambda(n)}{\log n}.$$
 * $$e^{\theta(x)}=\prod_{p\le x}p.$$
 * $$e^{\psi(x)}= \operatorname{lcm}[1,2,\dots,\lfloor x\rfloor].$$

Menon's identity
In 1965 P Kesava Menon proved $$\sum_{\stackrel{1\le k\le n}{ \gcd(k,n)=1}} \gcd(k-1,n)=\varphi(n)d(n).$$

This has been generalized by a number of mathematicians. For example, \sum_{\stackrel{1\le k_1, k_2, \dots, k_s\le n}{ \gcd(k_1,n)=1}} \gcd(k_1-1,k_2,\dots,k_s,n) = \varphi(n)\sigma_{s-1}(n).$$ \sum_{\stackrel{1\le k_1, k_2, \dots, k_s\le n}{ \gcd(k_1,k_2,\dots,k_s,n)=1}} \gcd(k_1-a_1,k_2-a_2,\dots,k_s-a_s,n)^s =J_s(n)d(n), $$ where a1, a2, ..., as are integers, gcd(a1, a2, ..., as, n) = 1. \sum_{\stackrel{1\le k\le m}{ \gcd(k,m)=1}} \gcd(k^2-1,m_1)\gcd(k^2-1,m_2) =\varphi(n)\sum_{\stackrel{d_1\mid m_1} {d_2\mid m_2}} \varphi(\gcd(d_1, d_2))2^{\omega(\operatorname{lcm}(d_1, d_2))}, $$ where m1 and m2 are odd, m = lcm(m1, m2).
 * B. Sury $$
 * N. Rao $$
 * László Fejes Tóth $$

In fact, if f is any arithmetical function $$\sum_{\stackrel{1\le k\le n}{ \gcd(k,n)=1}} f(\gcd(k-1,n)) =\varphi(n)\sum_{d\mid n}\frac{(\mu*f)(d)}{\varphi(d)},$$ 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: $$ \left(\frac{m}{n}\right) \left(\frac{n}{m}\right) = (-1)^{(m-1)(n-1)/4}.$$

Let D(n) be the arithmetic derivative. Then the logarithmic derivative $$\frac{D(n)}{n} = \sum_{\stackrel{p\mid n}{p\text{ prime}}} \frac {v_{p}(n)} {p}.$$ See Arithmetic derivative for details.

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

 * $$|\lambda(n)|\mu(n) =\lambda(n)|\mu(n)| = \mu(n),$$    and
 * $$\lambda(n)\mu(n) = |\mu(n)| =\mu^2(n).$$

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

 * $$\lambda(n)\mid \phi(n).$$      Further,

 * $$\lambda(n)= \phi(n) \text{ if and only if }n=\begin{cases}

1,2, 4;\\ 3,5,7,9,11, \ldots \text{ (that is, } p^k \text{, where }p\text{ is an odd prime)};\\ 6,10,14,18,\ldots \text{ (that is, } 2p^k\text{, where }p\text{ is an odd prime)}. \end{cases}$$ See Multiplicative group of integers modulo n and Primitive root modulo n.
 * $$2^{\omega(n)} \le d(n) \le 2^{\Omega(n)}.$$

 * $$\frac{6}{\pi^2}<\frac{\phi(n)\sigma(n)}{n^2} < 1.$$

 * $$\begin{align}

c_q(n) &=\frac{\mu\left(\frac{q}{\gcd(q, n)}\right)}{\phi\left(\frac{q}{\gcd(q, n)}\right)}\phi(q)\\ &=\sum_{\delta\mid \gcd(q,n)}\mu\left(\frac{q}{\delta}\right)\delta. \end{align}$$       Note that  $$\phi(q) = \sum_{\delta\mid q}\mu\left(\frac{q}{\delta}\right)\delta.$$

 * $$c_q(1) = \mu(q).$$
 * $$c_q(q) = \phi(q).$$

 * $$\sum_{\delta\mid n}d^{3}(\delta) = \left(\sum_{\delta\mid n}d(\delta)\right)^2.$$     Compare this with $1^{3} + 2^{3} + 3^{3} + ... + n^{3} = (1 + 2 + 3 + ... + n)^{2}$

 * $$d(uv) = \sum_{\delta\mid \gcd(u,v)}\mu(\delta)d\left(\frac{u}{\delta}\right)d\left(\frac{v}{\delta}\right).

$$

 * $$\sigma_k(u)\sigma_k(v) = \sum_{\delta\mid \gcd(u,v)}\delta^k\sigma_k\left(\frac{uv}{\delta^2}\right).

$$

 * $$\tau(u)\tau(v) = \sum_{\delta\mid \gcd(u,v)}\delta^{11}\tau\left(\frac{uv}{\delta^2}\right),

$$     where τ(n) is Ramanujan's function.

title: Arithmetic function


text:
ANSI C

ANSI C, ISO C, and Standard C are successive standards for the C programming language published by the American National Standards Institute (ANSI) and ISO/IEC JTC 1/SC 22/WG 14 of the International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC). Historically, the names referred specifically to the original and best-supported version of the standard (known as C89 or C90). Software developers writing in C are encouraged to conform to the standards, as doing so helps portability between compilers.

History and outlook
The first standard for C was published by ANSI. Although this document was subsequently adopted by ISO/IEC and subsequent revisions published by ISO/IEC have been adopted by ANSI, "ANSI C" is still used to refer to the standard. While some software developers use the term ISO C, others are standards-body neutral and use Standard C.

Informal specification: K&R C (C78)
Informal specification in 1978 (Brian Kernighan and Dennis Ritchie book The C Programming Language).

Standardizing C
In 1983, the American National Standards Institute formed a committee, X3J11, to establish a standard specification of C. In 1985, the first Standard Draft was released, sometimes referred to as C85. In 1986, another Draft Standard was released, sometimes referred to as C86. The prerelease Standard C was published in 1988, and sometimes referred to as C88.

C89
The ANSI standard was completed in 1989 and ratified as ANSI X3.159-1989 "Programming Language C." This version of the language is often referred to as "ANSI C". Later on sometimes the label "C89" is used to distinguish it from C90 but using the same labeling method.

C90
The same standard as C89 was ratified by ISO/IEC as ISO/IEC 9899:1990, with only formatting changes, which is sometimes referred to as C90. Therefore, the terms "C89" and "C90" refer to essentially the same language.

This standard has been withdrawn by both ANSI/INCITS and ISO/IEC.

C95
In 1995, the ISO/IEC published an extension, called Amendment 1, for the C standard. Its full name finally was ISO/IEC 9899:1990/AMD1:1995 or nicknamed C95. Aside from error correction there were further changes to the language capabilities, such as:

 * Improved multi-byte and wide character support in the standard library, introducing  and   as well as multi-byte I/O
 * Addition of digraphs to the language
 * Specification of standard macros for the alternative specification of operators, e.g.  for
 * Specification of the standard macro

In addition to the amendment, two technical corrigenda were published by ISO for C90:

 * ISO/IEC 9899:1990/Cor 1:1994 TCOR1 in 1994
 * ISO/IEC 9899:1990/Cor 2:1996 in 1996

C99
In March 2000, ANSI adopted the ISO/IEC 9899:1999 standard. This standard is commonly referred to as C99. Some notable additions to the previous standard include:

 * New built-in data types:,  ,  , and
 * Several new core language features, including static array indices, designated initializers, compound literals, variable-length arrays, flexible array members, variadic macros, and  keyword
 * Several new library headers, including,  ,  ,
 * Improved compatibility with several C++ features, including inline functions, single-line comments with, mixing declarations and code, and universal character names in identifiers
 * Removed several dangerous C89 language features such as implicit function declarations and implicit

Three technical corrigenda were published by ISO for C99:

 * ISO/IEC 9899:1999/Cor 1:2001(E)
 * ISO/IEC 9899:1999/Cor 2:2004(E)
 * ISO/IEC 9899:1999/Cor 3:2007(E), notable for deprecating the standard library function gets

This standard has been withdrawn by both ANSI/INCITS and ISO/IEC in favour of C11.

C11
C11 was officially ratified and published on December 8, 2011. Notable features include improved Unicode support, type-generic expressions using the new  keyword, a cross-platform multi-threading API, and atomic types support in both core language and the library.

One technical corrigendum has been published by ISO for C11:

 * ISO/IEC 9899:2011/Cor 1:2012

C17
, "C17" is the current standard for the C programming language.

C17 addresses defects in C11 without introducing new language features.

C23
C23 is the informal name for the next major C language standard revision.

Other related ISO publications
As part of the standardization process, ISO/IEC also publishes technical reports and specifications related to the C language:

More technical specifications are in development and pending approval, including the fifth and final part of TS 18661, a software transactional memory specification, and parallel library extensions.
 * ISO/IEC TR 19769:2004, on library extensions to support Unicode transformation formats, integrated into C11
 * ISO/IEC TR 24731-1:2007, on library extensions to support bounds-checked interfaces, integrated into C11
 * ISO/IEC TR 18037:2008, on embedded C extensions
 * ISO/IEC TR 24732:2009, on decimal floating point arithmetic, superseded by ISO/IEC TS 18661-2:2015
 * ISO/IEC TR 24747:2009, on special mathematical functions,
 * ISO/IEC TR 24731-2:2010, on library extensions to support dynamic allocation functions
 * ISO/IEC TS 17961:2013, on secure coding in C
 * ISO/IEC TS 18661-1:2014, on IEC 60559:2011-compatible binary floating-point arithmetic
 * ISO/IEC TS 18661-2:2015, on IEC 60559:2011-compatible decimal floating point arithmetic
 * ISO/IEC TS 18661-3:2015, on IEC 60559:2011-compatible interchange and extended floating-point types
 * ISO/IEC TS 18661-4:2015, on IEC 60559:2011-compatible supplementary functions

Support from major compilers
ANSI C is now supported by almost all the widely used compilers. GCC and Clang are two major C compilers popular today, both based on the C11 with updates including changes from later specifications such as C17. Any source code written only in standard C and without any hardware dependent assumptions is virtually guaranteed to compile correctly on any platform with a conforming C implementation. Without such precautions, most programs may compile only on a certain platform or with a particular compiler, due, for example, to the use of non-standard libraries, such as GUI libraries, or to the reliance on compiler- or platform-specific attributes such as the exact size of certain data types and byte endianness.

Compliance detectability
To mitigate the differences between K&R C and the ANSI C standard, the  ("standard c") macro can be used to split code into ANSI and K&R sections.

In the above example, a prototype is used in a function declaration for ANSI compliant implementations, while an obsolescent non-prototype declaration is used otherwise. Those are still ANSI-compliant as of C99. Note how this code checks both definition and evaluation: this is because some implementations may set  to zero to indicate non-ANSI compliance.

Compiler support
List of compilers supporting ANSI C:
 * Acornsoft ANSI C (first version in 1988, revised in 1989)
 * Amsterdam Compiler Kit (C K&R and C89/90)
 * ARM RealView
 * Clang, using LLVM backend
 * GCC (full C89/90, C99 and C11)
 * HP C/ANSI C compiler (C89 and C99)
 * IBM XL C/C++ (C11, starting with version 12.1)
 * Intel's ICC
 * LabWindows/CVI
 * LCC
 * Oracle Developer Studio
 * OpenWatcom (C89/90 and some C99)
 * Microsoft Visual C++ (C89/90 and some C99)
 * Pelles C (C99 and C11. Windows only.)
 * vbcc (C89/90 and C99)
 * Tiny C Compiler (C89/90 and some C99)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: ANSI C


text:
Alien and Sedition Acts

The Alien and Sedition Acts were a set of four laws enacted in 1798 that applied restrictions to immigration and speech in the United States. The Naturalization Act increased the requirements to seek citizenship, the Alien Friends Act allowed the president to imprison and deport non-citizens, the Alien Enemies Act gave the president additional powers to detain non-citizens during times of war, and the Sedition Act criminalized false and malicious statements about the federal government. The Alien Friends Act and the Sedition Act expired after a set number of years, and the Naturalization Act was repealed in 1802. The Alien Enemies Act is still in effect.

The Alien and Sedition Acts were controversial. They were supported by the Federalist Party, and supporters argued that the bills strengthened national security during the Quasi-War, an undeclared naval war with France from 1798 to 1800. The acts were denounced by Democratic-Republicans as suppression of voters and violation of free speech under the First Amendment. While they were in effect, the Alien and Sedition Acts, and the Sedition Act in particular, were used to suppress publishers affiliated with the Democratic-Republicans, and several publishers were arrested for criticism of the Adams administration. The Democratic-Republicans took power in 1800, because of backlash to the Alien and Sedition Acts, and all but the Alien Enemies Act were eliminated by the next Congress. The Alien Enemies Act has been invoked several times since, particularly during World War II. The Alien and Sedition Acts are generally received negatively by modern historians, and the U.S. Supreme Court has since indicated that aspects of the laws would likely be found unconstitutional today.

Alien Friends Act
The Alien Friends Act (officially "An Act Concerning Aliens") authorized the president to arbitrarily deport any non-citizen that was determined to be "dangerous to the peace and safety of the United States." Once a non-citizen was determined to be dangerous, or was suspected of conspiring against the government, the president had the power to set a reasonable amount of time for departure, and remaining after the time limit could result to up to three years in prison. The law was never directly enforced, but it was often used in conjunction with the Sedition Act to suppress criticism of the Adams administration. Upon enactment, the Alien Friends Act was authorized for two years, and it was allowed to expire at the end of this period. Democratic-Republicans opposed the law, with Thomas Jefferson referring to it as "a most detestable thing... worthy of the 8th or 9th century."

While the law was not directly enforced, it resulted in the voluntary departure of foreigners who feared that they would be charged under the act. The Adams administration encouraged these departures, and Secretary of State Timothy Pickering would ensure that the ships were granted passage. Though Adams did not delegate the final decision-making power, Secretary Pickering was responsible for overseeing enforcement of the Alien Friends Act. Both Adams and Pickering considered the law too weak to be effective; Pickering expressed his desire for the law to require sureties and authorize detainment prior to deportation.

Many French nationals were considered for deportation, but were allowed to leave willingly, or Adams declined to take action against them. These figures included: philosopher Constantin François de Chassebœuf, comte de Volney, General Victor Collot, scholar Médéric Louis Élie Moreau de Saint-Méry, diplomat Victor Marie du Pont, journalist William Duane, scientist Joseph Priestley, and journalist William Cobbett. Secretary Pickering also proposed applying the act against the French diplomatic delegation to the United States, but Adams refused. Journalist John Daly Burk agreed to leave under the act informally to avoid being tried for sedition, but he went into hiding in Virginia until the act's expiration.

Alien Enemies Act
The Alien Enemies Act (officially "An Act Respecting Alien Enemies") was passed to supplement the Alien Friends Act, granting the government additional powers to regulate non-citizens that would take effect in times of war. Under this law, the president could authorize the arrest, relocation, or deportation of any male over the age of 14 who hailed from a foreign enemy country. It also provided some legal protections for those subject to the law. Unlike the other acts, this act was largely unopposed by the Democratic-Republicans.

The Alien Enemies Act was not allowed to expire with the other Alien and Sedition Acts, and it remains in effect as Chapter 3, Sections 21–24 of Title 50 of the United States Code. President James Madison invoked the act against British nationals during the War of 1812. President Woodrow Wilson invoked the act against nationals of the Central Powers during World War I. In 1918, an amendment to the act struck the provision restricting the law to males.

On December 7, 1941, in response to the bombing of Pearl Harbor, President Franklin D. Roosevelt used the authority of the revised Alien Enemies Act to issue presidential proclamations #2525 (Alien Enemies – Japanese), #2526 (Alien Enemies – German), and #2527 (Alien Enemies – Italian), to apprehend, restrain, secure, and remove Japanese, German, and Italian non-citizens. Roosevelt later cited further wartime powers to issue Executive Order 9066, which interned Japanese Americans using powers unrelated to the Alien Enemies Act. Hostilities with Germany and Italy ended in May 1945, and President Harry S. Truman issued presidential proclamation #2655 on July 14. The proclamation gave the Attorney General authority regarding enemy aliens within the continental United States, to decide whether they are "dangerous to the public peace and safety of the United States," to order them removed, and to create regulations governing their removal, citing the Alien Enemies Act. On September 8, 1945, Truman issued presidential proclamation #2662, which authorized the Secretary of State to remove enemy aliens that had been sent to the United States from Latin American countries. On April 10, 1946, Truman issued presidential proclamation #2685, which modified the previous proclamation, and set a 30-day deadline for removal.

In Ludecke v. Watkins (1948), the Supreme Court interpreted the time of release under the Alien Enemies Act. German alien Kurt G. W. Ludecke was detained in 1941, under Proclamation 2526, and continued to be held after cessation of hostilities. In 1947, Ludecke petitioned for a writ of habeas corpus to order his release, after the Attorney General ordered him deported. The court ruled 5–4 to release Ludecke, but also found that the Alien Enemies Act allowed for detainment beyond the time hostilities ceased, until an actual treaty was signed with the hostile nation or government.

Naturalization Act
The Naturalization Act increased the residency requirement for American citizenship from five to 14 years, and increased the notice time from three to five years. At the time, the majority of immigrants supported Thomas Jefferson and the Democratic-Republicans—the political opponents of the Federalists. It did not have an expiration date, but it was repealed by the Naturalization Law of 1802.

Sedition Act
The Federalist-controlled Congress passed the Sedition Act by a vote of 44 to 41. The Sedition Act made it illegal to make false or malicious statements about the federal government. The act was used to suppress speech critical of the Adams administration, including the prosecution and conviction of many Jeffersonian newspaper owners who disagreed with the Federalist Party. The Sedition Act did not extend enforcement to speech about the Vice President, as then-incumbent Thomas Jefferson was a political opponent of the Federalist-controlled Congress. The Sedition Act was allowed to expire in 1800, and its enactment is credited with helping Jefferson win the presidential election that year.

Prominent prosecutions under the Sedition Act included:
 * James Thomson Callender, a British citizen, had been expelled from Great Britain for his political writings. Living first in Philadelphia, then seeking refuge close by in Virginia, he wrote a book titled The Prospect Before Us (read and approved by Vice President Jefferson before publication), in which he called the Adams administration a "continual tempest of malignant passions," and referred to the President as a "repulsive pedant, a gross hypocrite, and an unprincipled oppressor." Callender, already residing in Virginia and writing for the Richmond Examiner, was indicted in mid-1800 under the Sedition Act, and was subsequently convicted, fined $200, and sentenced to nine months in jail.
 * Matthew Lyon was a Democratic-Republican congressman from Vermont. He was the first individual to be placed on trial under the Alien and Sedition Acts. He was indicted in 1800 for an essay he had written in the Vermont Journal, where he had accused the administration of "ridiculous pomp, foolish adulation, and selfish avarice." While awaiting trial, Lyon commenced publication of Lyon's Republican Magazine, subtitled "The Scourge of Aristocracy." At trial, he was fined $1,000, and sentenced to four months in jail. After his release, he returned to Congress.
 * Benjamin Franklin Bache was the editor of the Philadelphia Aurora, a Democratic-Republican newspaper. Bache had accused George Washington of incompetence and financial irregularities, and "the blind, bald, crippled, toothless, querulous Adams" of nepotism and monarchical ambition. He was arrested in 1798 under the Sedition Act, but he died of yellow fever before trial.
 * Anthony Haswell was an English immigrant, and a printer of the Jeffersonian Vermont Gazette. Sourced from the Philadelphia Aurora, Haswell had reprinted Bache's claim that the federal government employed Tories. Haswell also published an advertisement from Lyon's sons for a lottery to raise money for his fine that decried Lyon's oppression by jailers exercising "usurped powers". Haswell was found guilty of seditious libel by judge William Paterson, and sentenced to a two-month imprisonment and a $200 fine.
 * Luther Baldwin was indicted, convicted, and fined $100 for a drunken incident that occurred during a visit by President Adams to Newark, New Jersey. Upon hearing a gun report during a parade, he yelled "I hope it hit Adams in the arse."
 * In November 1798, David Brown led a group in Dedham, Massachusetts, including Benjamin Fairbanks, in setting up a liberty pole with the words, "No Stamp Act, No Sedition Act, No Alien Bills, No Land Tax, downfall to the Tyrants of America; peace and retirement to the President; Long Live the Vice President." Brown was arrested in Andover, Massachusetts, but because he could not afford the $4,000 bail, he was taken to Salem for trial. Brown was tried in June 1799. Brown pleaded guilty, but Justice Samuel Chase asked him to name others who had assisted him. Brown refused, was fined $480, and sentenced to eighteen months in prison, the most severe sentence imposed under the Sedition Act.

History
The Alien and Sedition Acts were passed by Congress while it was controlled by the Federalist Party in 1798. Members of the Federalist Party grew increasingly distrustful of the opposing Democratic-Republican Party with the latter's support of France in the midst of the French Revolution. Some appeared to desire a similar revolution in the United States to overthrow the government and social structure. Newspapers sympathizing with each side exacerbated the tensions by accusing the other side's leaders of corruption, incompetence, and treason. The spreading unrest in Europe and calls for secession in the United States appeared to threaten the newly formed American republic. Some of this agitation was seen by Federalists as having been caused by French and French-sympathizing immigrants. The Alien and Sedition Acts were supported for different reasons, including the prevention of potential unrest by targeting immigrants, restriction of speech that may induce crime, reduction of partisan divides by penalizing expressly partisan speech, and suppression of political opponents of the Federalists.

The Acts were highly controversial at the time, especially the Sedition Act. The Sedition Act, which was signed into law by Adams on July 14, 1798, was hotly debated in the Federalist-controlled Congress and passed only after multiple amendments softening its terms, such as enabling defendants to argue in their defense that their statements had been true. Still, it passed the House only after three votes and another amendment causing it to automatically expire in March 1801. They continued to be loudly protested and were a major political issue in the election of 1800. Opposition to them resulted in the also-controversial Virginia and Kentucky Resolutions, authored by James Madison and Thomas Jefferson. Upon assuming the presidency, Thomas Jefferson pardoned those still serving sentences under the Sedition Act, and Congress soon repaid their fines.

Reaction
After the passage of the highly unpopular Alien and Sedition Acts, protests occurred across the country, with some of the largest being seen in Kentucky, where the crowds were so large they filled the streets and the entire town square of Lexington. Critics argued that they were primarily an attempt to suppress voters who disagreed with the Federalist party and its teachings, and violated the right of freedom of speech in the First Amendment to the U.S. Constitution. They also raised concerns that the Alien and Sedition acts gave disproportionate power to the federal executive compared to state governments and other branches of the federal government. Noting the outrage among the populace, the Democratic-Republicans made the Alien and Sedition Acts an important issue in the 1800 presidential election campaign. While government authorities prepared lists of aliens for deportation, many aliens fled the country during the debate over the Alien and Sedition Acts, and Adams never signed a deportation order.

The Virginia and Kentucky state legislatures also passed the Kentucky and Virginia Resolutions, secretly authored by Thomas Jefferson and James Madison, denouncing the federal legislation. While the eventual resolutions followed Madison in advocating "interposition", Jefferson's initial draft would have nullified the Acts and even threatened secession. Jefferson's biographer Dumas Malone argued that this might have gotten Jefferson impeached for treason, had his actions become known at the time. In writing the Kentucky Resolutions, Jefferson warned that, "unless arrested at the threshold", the Alien and Sedition Acts would "necessarily drive these states into revolution and blood".

The Alien and Sedition Acts were never appealed to the Supreme Court, whose power of judicial review was not established until Marbury v. Madison in 1803. Subsequent mentions in Supreme Court opinions beginning in the mid-20th century have assumed that the Sedition Act would today be found unconstitutional. Most modern historians view the Alien and Sedition Acts in a negative light, considering them to have been a mistake.

title: Alien and Sedition Acts


text:
Antinomy

Antinomy (Greek ἀντί, antí, "against, in opposition to", and νόμος, nómos, "law") refers to a real or apparent mutual incompatibility of two notions. It is a term used in logic and epistemology, particularly in the philosophy of Immanuel Kant.

There are many examples of antinomy. A self-contradictory phrase such as "There is no absolute truth" can be considered an antinomy because this statement is suggesting in itself to be an absolute truth, and therefore denies itself any truth in its statement. It is not necessarily also a paradox. A paradox, such as "this sentence is false" can also be considered to be an antinomy; in this case, for the sentence to be true, it must be false.

Kant's use
The term acquired a special significance in the philosophy of Immanuel Kant (1724–1804), who used it to describe the equally rational but contradictory results of applying to the universe of pure thought the categories or criteria of reason that are proper to the universe of sensible perception or experience (phenomena). Empirical reason cannot here play the role of establishing rational truths because it goes beyond possible experience and is applied to the sphere of that which transcends it.

For Kant there are four antinomies,  connected with:
 * the limitation of the universe in respect to space and time
 * the theory that the whole consists of indivisible atoms (whereas, in fact, none such exist)
 * the problem of free will in relation to universal causality
 * the existence of a universal being

In each antinomy, a thesis is contradicted by an antithesis. For example: in the first antinomy, Kant proves the thesis that time must have a beginning by showing that if time had no beginning, then an infinity would have elapsed up until the present moment. This is a manifest contradiction because infinity cannot, by definition, be completed by "successive synthesis"—yet just such a finalizing synthesis would be required by the view that time is infinite; so the thesis is proven. Then he proves the antithesis, that time has no beginning, by showing that if time had a beginning, then there must have been "empty time" out of which time arose. This is incoherent (for Kant) for the following reason: Since, necessarily, no time elapses in this pretemporal void, then there could be no alteration, and therefore nothing (including time) would ever come to be: so the antithesis is proven. Reason makes equal claim to each proof, since they are both correct, so the question of the limits of time must be regarded as meaningless.

This was part of Kant's critical program of determining limits to science and philosophical inquiry. These contradictions are inherent in reason when it is applied to the world as it is in itself, independently of any perception of it (this has to do with the distinction between phenomena and noumena). Kant's goal in his critical philosophy was to identify what claims are and are not justified, and the antinomies are a particularly illustrative example of his larger project.

Marx's use
In Das Kapital, Volume I in the chapter entitled "The Working Day", Karl Marx claims that capitalist production sustains "the assertion of a right to an unlimited working day, and the assertion of a right to a limited working day, both with equal justification". Furner emphasizes that the thesis and antithesis of this antinomy are not contradictory opposites, but rather "consist in the assertion of rights to states of affairs that are contradictory opposites".

title: Antinomy