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Basel problem

The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up more than a century later by Bernhard Riemann in his seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude", in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem.

The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series: $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots.$$

The sum of the series is approximately equal to 1.644934. The Basel problem asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be $$\pi^2/6$$ and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, although he was later proven correct. He produced an accepted proof in 1741.

The solution to this problem can be used to estimate the probability that two large random numbers are coprime. Two random integers in the range from 1 to $$n$$, in the limit as $$n$$ goes to infinity, are relatively prime with a probability that approaches $$6/\pi^2$$, the reciprocal of the solution to the Basel problem.

Euler's approach
Euler's original derivation of the value $$\pi^2/6$$ essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series.

Of course, Euler's original reasoning requires justification (100 years later, Karl Weierstrass proved that Euler's representation of the sine function as an infinite product is valid, by the Weierstrass factorization theorem), but even without justification, by simply obtaining the correct value, he was able to verify it numerically against partial sums of the series. The agreement he observed gave him sufficient confidence to announce his result to the mathematical community.

To follow Euler's argument, recall the Taylor series expansion of the sine function $$ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots $$ Dividing through by $$x$$ gives $$ \frac{\sin x}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots .$$

The Weierstrass factorization theorem shows that the left-hand side is the product of linear factors given by its roots, just as for finite polynomials. Euler assumed this as a heuristic for expanding an infinite degree polynomial in terms of its roots, but in fact it is not always true for general $$P(x)$$. This factorization expands the equation into: $$\begin{align} \frac{\sin x}{x} &= \left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 - \frac{x}{2\pi}\right)\left(1 + \frac{x}{2\pi}\right)\left(1 - \frac{x}{3\pi}\right)\left(1 + \frac{x}{3\pi}\right) \cdots \\ &= \left(1 - \frac{x^2}{\pi^2}\right)\left(1 - \frac{x^2}{4\pi^2}\right)\left(1 - \frac{x^2}{9\pi^2}\right) \cdots \end{align}$$

If we formally multiply out this product and collect all the $x^{2}$ terms (we are allowed to do so because of Newton's identities), we see by induction that the $x^{2}$ coefficient of $sin x⁄x$ is $$ -\left(\frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + \cdots \right) = -\frac{1}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2}.$$

But from the original infinite series expansion of $sin x⁄x$, the coefficient of $x^{2}$ is $−1⁄3! = −1⁄6$. These two coefficients must be equal; thus, $$-\frac{1}{6} = -\frac{1}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2}.$$

Multiplying both sides of this equation by −$\pi$2 gives the sum of the reciprocals of the positive square integers. $$\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}.$$

This method of calculating $$\zeta(2)$$ is detailed in expository fashion most notably in Havil's Gamma book which details many zeta function and logarithm-related series and integrals, as well as a historical perspective, related to the Euler gamma constant.

Generalizations of Euler's method using elementary symmetric polynomials
Using formulae obtained from elementary symmetric polynomials, this same approach can be used to enumerate formulae for the even-indexed even zeta constants which have the following known formula expanded by the Bernoulli numbers: $$\zeta(2n) = \frac{(-1)^{n-1} (2\pi)^{2n}}{2 \cdot (2n)!} B_{2n}. $$

For example, let the partial product for $$\sin(x)$$ expanded as above be defined by $$\frac{S_n(x)}{x} := \prod\limits_{k=1}^n \left(1 - \frac{x^2}{k^2 \cdot \pi^2}\right)$$. Then using known formulas for elementary symmetric polynomials (a.k.a., Newton's formulas expanded in terms of power sum identities), we can see (for example) that $$ \begin{align} \left[x^4\right] \frac{S_n(x)}{x} & = \frac{1}{2\pi^4}\left(\left(H_n^{(2)}\right)^2 - H_n^{(4)}\right) \qquad \xrightarrow{n \rightarrow \infty} \qquad \frac{1}{2\pi^4}\left(\zeta(2)^2-\zeta(4)\right) \\[4pt] & \qquad \implies \zeta(4) = \frac{\pi^4}{90} = -2\pi^4 \cdot [x^4] \frac{\sin(x)}{x} +\frac{\pi^4}{36} \\[8pt] \left[x^6\right] \frac{S_n(x)}{x} & = -\frac{1}{6\pi^6}\left(\left(H_n^{(2)}\right)^3 - 2H_n^{(2)} H_n^{(4)} + 2H_n^{(6)}\right) \qquad \xrightarrow{n \rightarrow \infty} \qquad \frac{1}{6\pi^6}\left(\zeta(2)^3-3\zeta(2)\zeta(4) + 2\zeta(6)\right) \\[4pt] & \qquad \implies \zeta(6) = \frac{\pi^6}{945} = -3 \cdot \pi^6 [x^6] \frac{\sin(x)}{x} - \frac{2}{3} \frac{\pi^2}{6} \frac{\pi^4}{90} + \frac{\pi^6}{216}, \end{align} $$

and so on for subsequent coefficients of $$[x^{2k}] \frac{S_n(x)}{x}$$. There are other forms of Newton's identities expressing the (finite) power sums $$H_n^{(2k)}$$ in terms of the elementary symmetric polynomials, $$e_i \equiv e_i\left(-\frac{\pi^2}{1^2}, -\frac{\pi^2}{2^2}, -\frac{\pi^2}{3^2}, -\frac{\pi^2}{4^2}, \ldots\right), $$ but we can go a more direct route to expressing non-recursive formulas for $$\zeta(2k)$$ using the method of elementary symmetric polynomials. Namely, we have a recurrence relation between the elementary symmetric polynomials and the power sum polynomials given as on this page by $$(-1)^{k}k e_k(x_1,\ldots,x_n) = \sum_{j=1}^k (-1)^{k-j-1} p_j(x_1,\ldots,x_n)e_{k-j}(x_1,\ldots,x_n),$$

which in our situation equates to the limiting recurrence relation (or generating function convolution, or product) expanded as $$ \frac{\pi^{2k}}{2}\cdot \frac{(2k) \cdot (-1)^k}{(2k+1)!} = -[x^{2k}] \frac{\sin(\pi x)}{\pi x} \times \sum_{i \geq 1} \zeta(2i) x^i. $$

Then by differentiation and rearrangement of the terms in the previous equation, we obtain that $$\zeta(2k) = [x^{2k}]\frac{1}{2}\left(1-\pi x\cot(\pi x)\right). $$

Consequences of Euler's proof
By the above results, we can conclude that $$\zeta(2k)$$ is always a rational multiple of $$\pi^{2k}$$. In particular, since $$\pi$$ and integer powers of it are transcendental, we can conclude at this point that $$\zeta(2k)$$ is irrational, and more precisely, transcendental for all $$k \geq 1$$. By contrast, the properties of the odd-indexed zeta constants, including Apéry's constant $$\zeta(3)$$, are almost completely unknown.

The Riemann zeta function
The Riemann zeta function $ζ(s)$ is one of the most significant functions in mathematics because of its relationship to the distribution of the prime numbers. The zeta function is defined for any complex number $s$ with real part greater than 1 by the following formula: $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.$$

Taking $s = 2$, we see that $ζ(2)$ is equal to the sum of the reciprocals of the squares of all positive integers: $$\zeta(2) = \sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6} \approx 1.644934.$$

Convergence can be proven by the integral test, or by the following inequality: $$\begin{align} \sum_{n=1}^N \frac{1}{n^2} & < 1 + \sum_{n=2}^N \frac{1}{n(n-1)} \\ & = 1 + \sum_{n=2}^N \left( \frac{1}{n-1} - \frac{1}{n} \right) \\ & = 1 + 1 - \frac{1}{N} \;{\stackrel{N \to \infty}{\longrightarrow}}\; 2. \end{align}$$

This gives us the upper bound 2, and because the infinite sum contains no negative terms, it must converge to a value strictly between 0 and 2. It can be shown that $ζ(s)$ has a simple expression in terms of the Bernoulli numbers whenever $s$ is a positive even integer. With $s = 2n$: $$\zeta(2n) = \frac{(2\pi)^{2n}(-1)^{n+1}B_{2n}}{2\cdot(2n)!}.$$

A proof using Euler's formula and L'Hôpital's rule
The normalized sinc function $$\text{sinc}(x)=\frac{\sin (\pi x)}{\pi x}$$ has a Weierstrass factorization representation as an infinite product: $$\frac{\sin (\pi x)}{\pi x} = \prod_{n=1}^\infty \left(1-\frac{x^2}{n^2}\right).$$

The infinite product is analytic, so taking the natural logarithm of both sides and differentiating yields $$\frac{\pi \cos (\pi x)}{\sin (\pi x)}-\frac{1}{x}=-\sum_{n=1}^\infty \frac{2x}{n^2-x^2}$$

(by uniform convergence, the interchange of the derivative and infinite series is permissible). After dividing the equation by $$2x$$ and regrouping one gets $$\frac{1}{2x^2}-\frac{\pi \cot (\pi x)}{2x}=\sum_{n=1}^\infty \frac{1}{n^2-x^2}.$$

We make a change of variables ($$x=-it$$): $$-\frac{1}{2t^2}+\frac{\pi \cot (-\pi it)}{2it}=\sum_{n=1}^\infty \frac{1}{n^2+t^2}.$$

Euler's formula can be used to deduce that $$\frac{\pi \cot (-\pi i t)}{2it}=\frac{\pi}{2it}\frac{i\left(e^{2\pi t}+1\right)}{e^{2\pi t}-1}=\frac{\pi}{2t}+\frac{\pi}{t\left(e^{2\pi t} - 1\right)}.$$ or using the corresponding hyperbolic function: $$\frac{\pi \cot (-\pi i t)}{2it}=\frac{\pi}{2t}{i\cot (\pi i t)}=\frac{\pi}{2t}\coth(\pi t).$$

Then $$\sum_{n=1}^\infty \frac{1}{n^2+t^2}=\frac{\pi \left(te^{2\pi t}+t\right)-e^{2\pi t}+1}{2\left(t^2 e^{2\pi t}-t^2\right)}=-\frac{1}{2t^2} + \frac{\pi}{2t} \coth(\pi t).$$

Now we take the limit as $$t$$ approaches zero and use L'Hôpital's rule thrice. By Tannery's theorem applied to $\lim_{t\to\infty}\sum_{n=1}^\infty 1/(n^2+1/t^2)$, we can interchange the limit and infinite series so that $\lim_{t\to 0}\sum_{n=1}^\infty 1/(n^2+t^2)=\sum_{n=1}^\infty 1/n^2$ and by L'Hôpital's rule $$\begin{align}\sum_{n=1}^\infty \frac{1}{n^2}&=\lim_{t\to 0}\frac{\pi}{4}\frac{2\pi te^{2\pi t}-e^{2\pi t}+1}{\pi t^2 e^{2\pi t} + te^{2\pi t}-t}\\[6pt] &=\lim_{t\to 0}\frac{\pi^3 te^{2\pi t}}{2\pi \left(\pi t^2 e^{2\pi t}+2te^{2\pi t} \right)+e^{2\pi t}-1}\\[6pt] &=\lim_{t\to 0}\frac{\pi^2 (2\pi t+1)}{4\pi^2 t^2+12\pi t+6}\\[6pt] &=\frac{\pi^2}{6}.\end{align}$$

A proof using Fourier series
Use Parseval's identity (applied to the function $f(x) = x$) to obtain $$\sum_{n=-\infty}^\infty |c_n|^2 = \frac{1}{2\pi}\int_{-\pi}^\pi x^2 \, dx,$$ where $$\begin{align} c_n &= \frac{1}{2\pi}\int_{-\pi}^\pi x e^{-inx} \, dx \\[4pt] &= \frac{n\pi \cos(n\pi)-\sin(n\pi)}{\pi n^2} i \\[4pt] &= \frac{\cos(n\pi)}{n} i \\[4pt] &= \frac{(-1)^n}{n} i \end{align}$$

for $n ≠ 0$, and $c_{0} = 0$. Thus, $$|c_n|^2 = \begin{cases} \dfrac{1}{n^2}, & \text{for } n \neq 0, \\ 0, & \text{for } n = 0, \end{cases} $$

and $$\sum_{n=-\infty}^\infty |c_n|^2 = 2\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{2\pi} \int_{-\pi}^\pi x^2 \, dx.$$

Therefore, $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{4\pi}\int_{-\pi}^\pi x^2 \, dx = \frac{\pi^2}{6}$$ as required.

Another proof using Parseval's identity
Given a complete orthonormal basis in the space $$L^2_{\operatorname{per}}(0, 1)$$ of L2 periodic functions over $$(0, 1)$$ (i.e., the subspace of square-integrable functions which are also periodic), denoted by $$\{e_i\}_{i=-\infty}^{\infty}$$, Parseval's identity tells us that $$\|x\|^2 = \sum_{i=-\infty}^{\infty} |\langle e_i, x\rangle|^2, $$

where $$\|x\| := \sqrt{\langle x,x\rangle}$$ is defined in terms of the inner product on this Hilbert space given by $$\langle f, g\rangle = \int_0^1 f(x) \overline{g(x)} \, dx,\ f,g \in L^2_{\operatorname{per}}(0, 1).$$

We can consider the orthonormal basis on this space defined by $$e_k \equiv e_k(\vartheta) := \exp(2\pi\imath k \vartheta)$$ such that $$\langle e_k,e_j\rangle = \int_0^1 e^{2\pi\imath (k-j) \vartheta} \, d\vartheta = \delta_{k,j}$$. Then if we take $$f(\vartheta) := \vartheta$$, we can compute both that $$ \begin{align} \|f\|^2 & = \int_0^1 \vartheta^2 \, d\vartheta = \frac{1}{3} \\ \langle f, e_k\rangle & = \int_0^1 \vartheta e^{-2\pi\imath k\vartheta} \, d\vartheta = \Biggl\{\begin{array}{ll} \frac{1}{2}, & k = 0 \\ -\frac{1}{2\pi\imath k} & k \neq 0, \end{array} \end{align} $$

by elementary calculus and integration by parts, respectively. Finally, by Parseval's identity stated in the form above, we obtain that $$ \begin{align} \|f\|^2 = \frac{1}{3} & = \sum_{\stackrel{k=-\infty}{k \neq 0}}^{\infty} \frac{1}{(2\pi k)^2}+ \frac{1}{4} = 2 \sum_{k=1}^{\infty} \frac{1}{(2\pi k)^2}+ \frac{1}{4} \\ & \implies \frac{\pi^2}{6} = \frac{2 \pi^2}{3} - \frac{\pi^2}{2} = \zeta(2). \end{align} $$

Generalizations and recurrence relations
Note that by considering higher-order powers of $$f_j(\vartheta) := \vartheta^j \in L^2_{\operatorname{per}}(0, 1)$$ we can use integration by parts to extend this method to enumerating formulas for $$\zeta(2j)$$ when $$j > 1$$. In particular, suppose we let $$I_{j,k} := \int_0^1 \vartheta^j e^{-2\pi\imath k\vartheta} \, d\vartheta, $$

so that integration by parts yields the recurrence relation that $$ \begin{align} I_{j,k} & = \begin{cases} \frac{1}{j+1}, & k=0; \\[4pt] -\frac{1}{2\pi\imath \cdot k} + \frac{j}{2\pi\imath \cdot k} I_{j-1,k}, & k \neq 0\end{cases} \\[6pt] & = \begin{cases} \frac{1}{j+1}, & k=0; \\[4pt] -\sum\limits_{m=1}^j \frac{j!}{(j+1-m)!} \cdot \frac{1}{(2\pi\imath \cdot k)^m}, & k \neq 0. \end{cases} \end{align} $$

Then by applying Parseval's identity as we did for the first case above along with the linearity of the inner product yields that $$ \begin{align} \|f_j\|^2 = \frac{1}{2j+1} & = 2 \sum_{k \geq 1} I_{j,k} \bar{I}_{j,k} + \frac{1}{(j+1)^2} \\[6pt] & = 2 \sum_{m=1}^j \sum_{r=1}^j \frac{j!^2}{(j+1-m)! (j+1-r)!} \frac{(-1)^r}{\imath^{m+r}} \frac{\zeta(m+r)}{(2\pi)^{m+r}} + \frac{1}{(j+1)^2}. \end{align} $$

Proof using differentiation under the integral sign
It's possible to prove the result using elementary calculus by applying the differentiation under the integral sign technique to an integral due to Freitas: $$I(\alpha) = \int_0^\infty \ln\left(1+\alpha e^{-x}+e^{-2x}\right)dx.$$

While the primitive function of the integrand cannot be expressed in terms of elementary functions, by differentiating with respect to $$\alpha$$ we arrive at

$$\frac{dI}{d\alpha} = \int_0^\infty \frac{e^{-x}}{1+\alpha e^{-x}+e^{-2x}}dx,$$ which can be integrated by substituting $$u=e^{-x}$$ and decomposing into partial fractions. In the range $$-2\leq\alpha\leq 2$$ the definite integral reduces to

$$\frac{dI}{d\alpha} = \frac{2}{\sqrt{4-\alpha^2}}\left[\arctan\left(\frac{\alpha+2}{\sqrt{4-\alpha^2}}\right)-\arctan\left(\frac{\alpha}{\sqrt{4-\alpha^2}}\right)\right].$$

The expression can be simplified using the arctangent addition formula and integrated with respect to $$\alpha$$ by means of trigonometric substitution, resulting in

$$I(\alpha) = -\frac{1}{2}\arccos\left(\frac{\alpha}{2}\right)^2 + c.$$

The integration constant $$c$$ can be determined by noticing that two distinct values of $$I(\alpha)$$ are related by

$$I(2) = 4I(0),$$ because when calculating $$I(2)$$ we can factor $$1+2e^{-x}+e^{-2x} = (1+e^{-x})^2$$ and express it in terms of $$I(0)$$ using the logarithm of a power identity and the substitution $$u=x/2$$. This makes it possible to determine $$c = \frac{\pi^2}{6}$$, and it follows that

$$I(-2) = 2\int_0^\infty \ln(1-e^{-x})dx = -\frac{\pi^2}{3}.$$

This final integral can be evaluated by expanding the natural logarithm into its Taylor series:

$$\int_0^\infty \ln(1-e^{-x})dx = - \sum_{n=1}^\infty \int_0^\infty \frac{e^{-nx}}{n}dx = -\sum_{n=1}^\infty\frac{1}{n^2}.$$

The last two identities imply

$$\sum_{n=1}^\infty\frac{1}{n^2} = \frac{\pi^2}{6}.$$

Cauchy's proof
While most proofs use results from advanced mathematics, such as Fourier analysis, complex analysis, and multivariable calculus, the following does not even require single-variable calculus (until a single limit is taken at the end).

For a proof using the residue theorem, see here.

History of this proof
The proof goes back to Augustin Louis Cauchy (Cours d'Analyse, 1821, Note VIII). In 1954, this proof appeared in the book of Akiva and Isaak Yaglom "Nonelementary Problems in an Elementary Exposition". Later, in 1982, it appeared in the journal Eureka, attributed to John Scholes, but Scholes claims he learned the proof from Peter Swinnerton-Dyer, and in any case he maintains the proof was "common knowledge at Cambridge in the late 1960s".

The proof
[[File:limit circle FbN.jpeg|thumb|The inequality $$\tfrac{1}{2}r^2\tan\theta > \tfrac{1}{2}r^2\theta > \tfrac{1}{2}r^2\sin\theta$$ is shown pictorially for any $$\theta \in (0, \pi/2)$$. The three terms are the areas of the triangle OAC, circle section OAB, and the triangle OAB.

Taking reciprocals and squaring gives $$\cot^2\theta<\tfrac{1}{\theta^2}<\csc^2\theta$$.]] The main idea behind the proof is to bound the partial (finite) sums $$\sum_{k=1}^m \frac{1}{k^2} = \frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{m^2}$$ between two expressions, each of which will tend to $\pi^{2}⁄6$ as $m$ approaches infinity. The two expressions are derived from identities involving the cotangent and cosecant functions. These identities are in turn derived from de Moivre's formula, and we now turn to establishing these identities.

Let $x$ be a real number with $0 < x < \pi⁄2$, and let $n$ be a positive odd integer. Then from de Moivre's formula and the definition of the cotangent function, we have $$\begin{align} \frac{\cos (nx) + i \sin (nx)}{\sin^n x} &= \frac{(\cos x + i\sin x)^n}{\sin^n x} \\[4pt] &= \left(\frac{\cos x + i \sin x}{\sin x}\right)^n \\[4pt] &= (\cot x + i)^n. \end{align}$$

From the binomial theorem, we have $$\begin{align} (\cot x + i)^n = & {n \choose 0} \cot^n x + {n \choose 1} (\cot^{n - 1} x)i + \cdots + {n \choose {n - 1}} (\cot x)i^{n - 1} + {n \choose n} i^n \\[6pt] = & \Bigg( {n \choose 0} \cot^n x - {n \choose 2} \cot^{n - 2} x \pm \cdots \Bigg) \; + \; i\Bigg( {n \choose 1} \cot^{n-1} x - {n \choose 3} \cot^{n - 3} x \pm \cdots \Bigg). \end{align}$$

Combining the two equations and equating imaginary parts gives the identity $$\frac{\sin (nx)}{\sin^n x} = \Bigg( {n \choose 1} \cot^{n - 1} x - {n \choose 3} \cot^{n - 3} x \pm \cdots \Bigg).$$

We take this identity, fix a positive integer $m$, set $n = 2m + 1$, and consider $x_{r} = r\pi⁄2m + 1$ for $r = 1, 2, ..., m$. Then $nx_{r}$ is a multiple of π and therefore $sin(nx_{r}) = 0$. So, $$0 = {{2m + 1} \choose 1} \cot^{2m} x_r - {{2m + 1} \choose 3} \cot^{2m - 2} x_r \pm \cdots + (-1)^m{{2m + 1} \choose {2m + 1}}$$

for every $r = 1, 2, ..., m$. The values $x_{r} = x_{1}, x_{2}, ..., x_{m}$ are distinct numbers in the interval $0 <

x_{r}

< \pi⁄2$. Since the function $cot^{2} x$ is one-to-one on this interval, the numbers $t_{r} = cot^{2} x_{r}$ are distinct for $r = 1, 2, ..., m$. By the above equation, these $m$ numbers are the roots of the $m$th degree polynomial $$p(t) = {{2m + 1} \choose 1}t^m - {{2m + 1} \choose 3}t^{m - 1} \pm \cdots + (-1)^m{{2m+1} \choose {2m + 1}}.$$

By Vieta's formulas we can calculate the sum of the roots directly by examining the first two coefficients of the polynomial, and this comparison shows that $$\cot ^2 x_1 + \cot ^2 x_2 + \cdots + \cot ^2 x_m = \frac{\binom{2m + 1}3} {\binom{2m + 1}1} = \frac{2m(2m - 1)}6.$$

Substituting the identity $csc^{2} x = cot^{2} x + 1$, we have $$\csc ^2 x_1 + \csc ^2 x_2 + \cdots + \csc ^2 x_m = \frac{2m(2m - 1)}6 + m = \frac{2m(2m + 2)}6.$$

Now consider the inequality $cot^{2} x < 1⁄x^{2} < csc^{2} x$ (illustrated geometrically above). If we add up all these inequalities for each of the numbers $x_{r} = r\pi⁄2m + 1$, and if we use the two identities above, we get $$\frac{2m(2m - 1)}6 < \left(\frac{2m + 1}{\pi} \right)^2 + \left(\frac{2m + 1}{2\pi} \right)^2 + \cdots + \left(\frac{2m + 1}{m \pi} \right)^2 < \frac{2m(2m + 2)}6.$$

Multiplying through by $( \pi⁄2m + 1 )  2$, this becomes $$\frac{\pi ^2}{6}\left(\frac{2m}{2m + 1}\right)\left(\frac{2m - 1}{2m + 1}\right) < \frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{m^2} < \frac{\pi ^2}{6}\left(\frac{2m}{2m + 1}\right)\left(\frac{2m + 2}{2m + 1}\right).$$

As $m$ approaches infinity, the left and right hand expressions each approach $\pi^{2}⁄6$, so by the squeeze theorem, $$\zeta(2) = \sum_{k=1}^\infty \frac{1}{k^2} = \lim_{m \to \infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{m^2}\right) = \frac{\pi ^2}{6}$$

and this completes the proof.

Proof assuming Weil's conjecture on Tamagawa numbers
A proof is also possible assuming Weil's conjecture on Tamagawa numbers. The conjecture asserts for the case of the algebraic group SL2(R) that the Tamagawa number of the group is one. That is, the quotient of the special linear group over the rational adeles by the special linear group of the rationals (a compact set, because $$SL_2(\mathbb Q)$$ is a lattice in the adeles) has Tamagawa measure 1: $$\tau(SL_2(\mathbb Q)\setminus SL_2(A_{\mathbb Q}))=1.$$

To determine a Tamagawa measure, the group $$SL_2$$ consists of matrices $$\begin{bmatrix}x&y\\z&t\end{bmatrix}$$ with $$xt-yz=1$$. An invariant volume form on the group is $$\omega = \frac1x dx\wedge dy\wedge dz.$$

The measure of the quotient is the product of the measures of $$SL_2(\mathbb Z)\setminus SL_2(\mathbb R)$$ corresponding to the infinite place, and the measures of $$SL_2(\mathbb Z_p)$$ in each finite place, where $$\mathbb Z_p$$ is the p-adic integers.

For the local factors, $$\omega(SL_2(\mathbb Z_p)) = |SL_2(F_p)|\omega(SL_2(\mathbb Z_p,p))$$ where $$F_p$$ is the field with $$p$$ elements, and $$SL_2(\mathbb Z_p,p)$$ is the congruence subgroup modulo $$p$$. Since each of the coordinates $$x,y,z$$ map the latter group onto $$p\mathbb Z_p$$ and $$\left|\frac1x\right|_p=1$$, the measure of $$SL_2(\mathbb Z_p,p)$$ is $$\mu_p(p\mathbb Z_p)^3=p^{-3}$$, where $$\mu_p$$ is the normalized Haar measure on $$\mathbb Z_p$$. Also, a standard computation shows that $$|SL_2(F_p)|=p(p^2-1)$$. Putting these together gives $$\omega(SL_2(\mathbb Z_p))=(1-1/p^2)$$.

At the infinite place, an integral computation over the fundamental domain of $$SL_2(\mathbb Z)$$ shows that $$\omega(SL_2(\mathbb Z)\setminus SL_2(\mathbb R)=\pi^2/6$$, and therefore the Weil conjecture finally gives $$1 = \frac{\pi^2}6\prod_p \left(1-\frac1{p^2}\right).$$ On the right-hand side, we recognize the Euler product for $$1/\zeta(2)$$, and so this gives the solution to the Basel problem.

This approach shows the connection between (hyperbolic) geometry and arithmetic, and can be inverted to give a proof of the Weil conjecture for the special case of $$SL_2$$, contingent on an independent proof that $$\zeta(2)=\pi^2/6$$.

Other identities
See the special cases of the identities for the Riemann zeta function when $$s = 2.$$ Other notably special identities and representations of this constant appear in the sections below.

Series representations
The following are series representations of the constant: $$\begin{align} \zeta(2) &= 3 \sum_{k=1}^\infty \frac{1}{k^2 \binom{2k}{k}} \\[6pt] &= \sum_{i=1}^\infty \sum_{j=1}^\infty \frac{(i-1)! (j-1)!}{(i+j)!}. \end{align}$$

There are also BBP-type series expansions for $ζ(2)$.

Integral representations
The following are integral representations of $$\zeta(2)\text{:}$$ $$ \begin{align} \zeta(2) & = -\int_0^1 \frac{\log x}{1-x} \, dx \\[6pt] & = \int_0^{\infty} \frac{x}{e^x-1} \, dx \\[6pt] & = \int_0^1 \frac{(\log x)^2}{(1+x)^2} \, dx \\[6pt] & = 2 + 2\int_1^{\infty} \frac{\lfloor x \rfloor -x}{x^3} \, dx \\[6pt] & = \exp\left(2 \int_2^{\infty} \frac{\pi(x)}{x(x^2-1)} \,dx\right) \\[6pt] & = \int_0^1 \int_0^1 \frac{dx \, dy}{1-xy} \\[6pt] & = \frac{4}{3} \int_0^1 \int_0^1 \frac{dx \, dy}{1-(xy)^2} \\[6pt] & = \int_0^1 \int_0^1 \frac{1-x}{1-xy} \, dx \, dy + \frac{2}{3}. \end{align}$$

Continued fractions
In van der Poorten's classic article chronicling Apéry's proof of the irrationality of $\zeta(3)$, the author notes as "a red herring" the similarity of a continued fraction for Apery's constant, and the following one for the Basel constant: $$\frac{\zeta(2)}{5} = \cfrac{1}{\widetilde{v}_1 - \cfrac{1^4}{\widetilde{v}_2-\cfrac{2^4}{\widetilde{v}_3-\cfrac{3^4}{\widetilde{v}_4-\ddots}}}}, $$ where $$\widetilde{v}_n = 11n^2-11n+3 \mapsto \{3,25,69,135,\ldots\}$$. Another continued fraction of a similar form is: $$\frac{\zeta(2)}{2} = \cfrac{1}{v_1 - \cfrac{1^4}{v_2-\cfrac{2^4}{v_3-\cfrac{3^4}{v_4-\ddots}}}}, $$ where $$v_n = 2n-1 \mapsto \{1,3,5,7,9,\ldots\}$$.

title: Basel problem


text:
Guy Hamilton

Mervyn Ian Guy Hamilton, DSC (16 September 1922 – 20 April 2016) was an English film director. He directed 22 films from the 1950s to the 1980s, including four James Bond films.

Early life
Hamilton was born in Paris on 16 September 1922, son of Frederick William Guy Hamilton (1895-1988), press attaché to the British embassy in Paris and Captain in the King's Own Royal Lancaster Regiment, and Winifred Grace Culling (1895-1970), daughter of William Archibald Culling Fremantle, of the Church Missionary Society in India. His mother was a great-granddaughter of the Christian campaigner Sir Culling Eardley, 3rd Baronet, and of the politician Thomas Fremantle, 1st Baron Cottesloe. His parents divorced in 1923, and Hamilton attended school in England. His first exposure to the film industry came in 1938, when he was a clapperboard boy at the Victorine Studios in Nice. At the outbreak of the Second World War in 1939, Hamilton escaped from France by the MV Saltersgate, a collier bound for French North Africa; one of the other 500 refugees aboard was W. Somerset Maugham.

Having travelled from Oran to Gibraltar before arriving in London, he worked in the film library at Paramount News before being commissioned in the Royal Navy; he served in the 15th Motor Torpedo Boat 718 Flotilla, a unit that ferried agents into France and brought downed British pilots back to England.

During this service, he was left behind for a month in occupied Brittany; he was later awarded the Distinguished Service Cross.

Assistant Director
After the war Hamilton wanted to get into film production and managed to get a job on a second unit in Dartmoor on a Trevor Heid picture. Then he was put under contract by Alexander Korda as a third assistant director. Over the next few years he worked his way up to a first assistant director.

Hamilton later said, "I found that working with bad directors was infinitely more useful because you watched them get into trouble three times a day and puddle around and you say, you know, I won't do that, I don't want to fall into that trap."

He worked on They Made Me a Fugitive (1947), Mine Own Executioner (1947), Anna Karenina (1948), and The Fallen Idol (1949) directed by Carol Reed. "I was devoted to Carol," said Hamilton later. "He made my life easy because I followed him around like a little dog while learning my trade. If you’d ask him a question, he’d always answer it...Carol Reed was the biggest influence on me and on everything that I did."

Hamilton assisted on Britannia Mews (1949), a 20th Century Fox film shot in England, directed by Jean Negulesco; was reunited with Reed on The Third Man (1949), in which Hamilton doubled for Orson Welles in a couple of shots; The Angel with the Trumpet (1950), State Secret (1950) for Sidney Gilliat; Outcast of the Islands (1951) for Reed; The African Queen (1951) for John Huston; and Home at Seven (1952) for Ralph Richardson.

Early films as director
Reed suggested to Hamilton that if he wanted to direct he should refuse to re-sign with Korda unless he gave Hamilton a chance to direct. It worked and Korda allocated Hamilton the job of directing the B-movie The Ringer (1952).

Hamilton's second film as director was The Intruder (1953) dealing with soldiers returning to civilian life, produced by Ivan Foxwell. He followed it with an adaptation of An Inspector Calls (1954).

Hamilton's fourth film was the prisoner-of-war story The Colditz Story (1955), which he also co-wrote with producer Foxwell. It was his highest-grossing movie of the decade. He also tried a musical with Max Bygraves, Charley Moon (1956) and an adventure film which he co wrote with Foxwell, Manuela (1957).

Hamilton had his first experience with larger-budget films towards the end of the decade, when he replaced the sacked Alexander Mackendrick on the set of The Devil's Disciple (1959) featuring Kirk Douglas and Burt Lancaster.

After A Touch of Larceny (1960), which he co wrote with producer Foxwell, Hamilton again found himself working with a war theme on the Dino De Laurentiis-produced Italian war comedy The Best of Enemies (1961). This was the first film to show Hamilton's skill with intricate set-piece action sequences.

He turned down an offer to direct Dr. No (1962), the first James Bond film. His next release, and somewhat outside his developing œuvre, was The Party's Over, which, though filmed in 1963, was not released until 1965. The film was heavily censored and, in protest, Hamilton asked for his name to be removed when the film was finally released. He directed Man in the Middle (1964) with Robert Mitchum.

James Bond
Hamilton followed with his first James Bond film, Goldfinger (1964). He later reflected that he was able to successfully merge the series's mix of action adventure, sexual innuendo and black humour.

In the late 1960s, Hamilton directed two further films for Bond producer Harry Saltzman: Funeral in Berlin (1966, starring Michael Caine), and the war epic Battle of Britain (1969).

He returned to the Bond film franchise with the chase- and gadget-dependent Diamonds Are Forever (1971), Live and Let Die (1973) and The Man with the Golden Gun (1974). He claimed in a much later interview that he had instructed Roger Moore not to mimic Sean Connery's rendition of James Bond and said the only Bond he regretted making was Golden Gun. Hamilton was later asked to direct For Your Eyes Only (1981) but declined because the cash-strapped United Artists could not afford his salary.

Hamilton was originally chosen to direct Superman: The Movie (1978), but due to his status as a tax exile, he was allowed to be in England for only thirty days a year, where production had moved at the last minute to Pinewood Studios. The job of director was then passed to Richard Donner, but Hamilton insisted that he be paid in full.

Hamilton's only film projects in the latter part of the 1970s were the commercially unsuccessful Force 10 from Navarone (1978) and the poorly received adaptation of Agatha Christie's mystery The Mirror Crack'd (1980).

Another Christie adaptation followed in 1982, with Evil Under the Sun which was received more favourably than The Mirror Crack'd.

Hamilton directed only two more films in the 1980s (Remo Williams: The Adventure Begins in 1985 and 1989's Try This One for Size) before entering retirement.

In the late 1980s he was approached to direct Batman (1989), but declined. In a 2003 interview, he said that the contemporary Bond films relied too heavily on special effects and not as much on the spectacular and risky stunts of the Bond films of his era.

"I know that I’ve made some bad pictures, but when I was making a film, I knew I had to do the best I could with the material that I was working with," he said. "Sometimes I wished I had a more cooperative or a better writer, but that's the same for everybody."

Death
Hamilton died at the age of 93 on 20 April 2016.

Personal life
Hamilton was married twice, first to Naomi Chance in 1953, and then to the actress Kerima in 1964, many years after they first met during the filming of Outcast of the Islands. They lived in a villa in Andratx on the Mediterranean island of Mallorca from the mid-1970s until his death.

James Bond

 * Goldfinger (1964)
 * Diamonds Are Forever (1971)
 * Live and Let Die (1973)
 * The Man with the Golden Gun (1974)

Other films

 * The Ringer (1952)
 * The Intruder (1953)
 * An Inspector Calls (1954)
 * The Colditz Story (1955) (also co-writer)
 * Charley Moon (1956)
 * Manuela (1957)
 * A Touch of Larceny (1959)
 * The Devil's Disciple (1959)
 * The Best of Enemies (1961)
 * Man in the Middle (1964)
 * The Party's Over (1965)
 * Funeral in Berlin (1966)
 * Battle of Britain (1969)
 * Force 10 from Navarone (1978)
 * The Mirror Crack'd (1980)
 * Evil Under the Sun (1982)
 * Remo Williams: The Adventure Begins (1985)
 * Try This One for Size (1989)

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

title: Guy Hamilton


text:
Ischemia

Ischemia or ischaemia is a restriction in blood supply to any tissue, muscle group, or organ of the body, causing a shortage of oxygen that is needed for cellular metabolism (to keep tissue alive). Ischemia is generally caused by problems with blood vessels, with resultant damage to or dysfunction of tissue i.e. hypoxia and microvascular dysfunction. It also implies local hypoxia in a part of a body resulting from constriction (such as vasoconstriction, thrombosis, or embolism).

Ischemia causes not only insufficiency of oxygen, but also reduced availability of nutrients and inadequate removal of metabolic wastes. Ischemia can be partial (poor perfusion) or total blockage. The inadequate delivery of oxygenated blood to the organs must be resolved either by treating the cause of the inadequate delivery or reducing the oxygen demand of the system that needs it. For example, patients with myocardial ischemia have a decreased blood flow to the heart and are prescribed with medications that reduce chronotrophy and ionotrophy to meet the new level of blood delivery supplied by the stenosed vasculature so that it is adequate.

Signs and symptoms
The signs and symptoms of ischemia vary, as they can occur anywhere in the body and depend on the degree to which blood flow is interrupted. For example, clinical manifestations of acute limb ischemia (which can be summarized as the "six P's") include pain, pallor, pulseless, paresthesia, paralysis, and poikilothermia.

Without immediate intervention, ischemia may progress quickly to tissue necrosis and gangrene within a few hours. Paralysis is a very late sign of acute arterial ischemia and signals the death of nerves supplying the extremity. Foot drop may occur as a result of nerve damage. Because nerves are extremely sensitive to hypoxia, limb paralysis or ischemic neuropathy may persist after revascularization and may be permanent.

Cardiac ischemia
Cardiac ischemia may be asymptomatic or may cause chest pain, known as angina pectoris. It occurs when the heart muscle, or myocardium, receives insufficient blood flow. This most frequently results from atherosclerosis, which is the long-term accumulation of cholesterol-rich plaques in the coronary arteries. In most Western countries, Ischemic heart disease is the most common cause of death in both men and women, and a major cause of hospital admissions.

Bowel
Both large and small intestines can be affected by ischemia. The blockage of blood flow to the large intestine (colon) is called ischemic colitis. Ischemia of the small bowel is called mesenteric ischemia.

Brain
Brain ischemia is insufficient blood flow to the brain, and can be acute or chronic. Acute ischemic stroke is a neurological emergency typically caused by a blood clot blocking blood flow in a vessel in the brain. Chronic ischemia of the brain may result in a form of dementia called vascular dementia. A sudden, brief episode (symptoms lasting only minutes) of ischemia affecting the brain is called a transient ischemic attack (TIA), often called a mini-stroke. TIAs can be a warning of future strokes, with approximately 1/3 of TIA patients having a serious stroke within one year.

Limb
Inadequate blood supply to a limb may result in acute limb ischemia or chronic limb threatening ischemia.

Cutaneous
Reduced blood flow to the skin layers may result in mottling or uneven, patchy discoloration of the skin.

Kidney ischemia
Kidney ischemia is a loss of blood flow to the kidney cells. Several physical symptoms include shrinkage of one or both kidneys, renovascular hypertension, acute renal failure, progressive azotemia, and acute pulmonary edema. It is a disease with high mortality rate and high morbidity. Failure to treat could cause chronic kidney disease and a need for renal surgery.

Causes
Ischemia is a vascular disease involving an interruption in the arterial blood supply to a tissue, organ, or extremity that, if untreated, can lead to tissue death. It can be caused by embolism, thrombosis of an atherosclerotic artery, or trauma. Venous problems like venous outflow obstruction and low-flow states can cause acute arterial ischemia. An aneurysm is one of the most frequent causes of acute arterial ischemia. Other causes are heart conditions including myocardial infarction, mitral valve disease, chronic atrial fibrillation, cardiomyopathies, and prosthesis, in all of which thrombi are prone to develop.

Occlusion
The thrombi may dislodge and may travel anywhere in the circulatory system, where they may lead to pulmonary embolus, an acute arterial occlusion causing the oxygen and blood supply distal to the embolus to decrease suddenly. The degree and extent of symptoms depend on the size and location of the obstruction, the occurrence of clot fragmentation with embolism to smaller vessels, and the degree of peripheral arterial disease (PAD).
 * Thromboembolism (blood clots)
 * Embolism (foreign bodies in the circulation, e.g. amniotic fluid embolism)

Trauma
Traumatic injury to an extremity may produce partial or total occlusion of a vessel from compression, shearing, or laceration. Acute arterial occlusion may develop as a result of arterial dissection in the carotid artery or aorta or as a result of iatrogenic arterial injury (e.g., after angiography).

Other
An inadequate flow of blood to a part of the body may be caused by any of the following:
 * Thoracic outlet syndrome (compression of the brachial plexus)
 * Atherosclerosis (lipid-laden plaques obstructing the lumen of arteries)
 * Hypoglycemia (lower than normal level of glucose)
 * Tachycardia (abnormally rapid beating of the heart)
 * Radiotherapy, therapeutic radiation used to treat cancer can cause a delayed side effect injury in adjacent tissue via progressive, proliferative endarteritis, inflamed arterial linings that disrupt the tissue's blood supply.
 * Hypotension (low blood pressure, e.g. in septic shock, heart failure)
 * Outside compression of a blood vessel, e.g. by a tumor or in the case of superior mesenteric artery syndrome
 * Sickle cell disease (abnormally shaped red blood cells)
 * Induced g-forces which restrict the blood flow and force the blood to the extremities of the body, as in acrobatics and military flying
 * Localized extreme cold, such as by frostbite or improper cold compression therapy
 * Tourniquet application
 * An increased level of glutamate receptor stimulation
 * Arteriovenous malformations and peripheral artery occlusive disease
 * rupture of significant blood vessels supplying a tissue or organ.
 * Anemia vasoconstricts the periphery so that red blood cells cannot work internally on vital organs such as the heart, brain, etc., thus causing lack of oxygen to the periphery.
 * Premature discontinuation of any oral anticoagulant.
 * Unconsciousness, such as due to the ingestion of excessive doses of central depressants like alcohol or opioids, can result in ischemia of the extremities due to unusual body positions that prevent normal circulation

Pathophysiology

Ischemia results in tissue damage in a process known as ischemic cascade. The damage is the result of the build-up of metabolic waste products, inability to maintain cell membranes, mitochondrial damage, and eventual leakage of autolyzing proteolytic enzymes into the cell and surrounding tissues.

Restoration of blood supply to ischemic tissues can cause additional damage known as reperfusion injury that can be more damaging than the initial ischemia. Reintroduction of blood flow brings oxygen back to the tissues, causing a greater production of free radicals and reactive oxygen species that damage cells. It also brings more calcium ions to the tissues causing further calcium overloading and can result in potentially fatal cardiac arrhythmias and also accelerates cellular self-destruction. The restored blood flow also exaggerates the inflammation response of damaged tissues, causing white blood cells to destroy damaged cells that may otherwise still be viable.

Treatment
Early treatment is essential to keep the affected organ viable. The treatment options include injection of an anticoagulant, thrombolysis, embolectomy, surgical revascularization, or partial amputation. Anticoagulant therapy is initiated to prevent further enlargement of the thrombus. Continuous IV unfractionated heparin has been the traditional agent of choice.

If the condition of the ischemic limb is stabilized with anticoagulation, recently formed emboli may be treated with catheter-directed thrombolysis using intra-arterial infusion of a thrombolytic agent (e.g., recombinant tissue plasminogen activator (tPA), streptokinase, or urokinase). A percutaneous catheter inserted into the femoral artery and threaded to the site of the clot is used to infuse the drug. Unlike anticoagulants, thrombolytic agents work directly to resolve the clot over a period of 24 to 48 hours.

Direct arteriotomy may be necessary to remove the clot. Surgical revascularization may be used in the setting of trauma (e.g., laceration of the artery). Amputation is reserved for cases where limb salvage is not possible. If the patient continues to have a risk of further embolization from some persistent source, such as chronic atrial fibrillation, treatment includes long-term oral anticoagulation to prevent further acute arterial ischemic episodes.

Decrease in body temperature reduces the aerobic metabolic rate of the affected cells, reducing the immediate effects of hypoxia. Reduction of body temperature also reduces the inflammation response and reperfusion injury. For frostbite injuries, limiting thawing and warming of tissues until warmer temperatures can be sustained may reduce reperfusion injury.

Ischemic stroke is at times treated with various levels of statin therapy at hospital discharge, followed by home time, in an attempt to lower the risk of adverse events.

Society and culture
The Infarct Combat Project (ICP) is an international nonprofit organization founded in 1998 to fight ischemic heart diseases through education and research.

Etymology and pronunciation
The word ischemia is from Greek ἴσχαιμος iskhaimos 'staunching blood', from ἴσχω iskhο 'keep back, restrain' and αἷμα haima 'blood'.

title: Ischemia


text:
121 (number)

121 (one hundred [and] twenty-one) is the natural number following 120 and preceding 122.

In mathematics
One hundred [and] twenty-one is
 * a square (11 times 11)
 * the sum of the powers of 3 from 0 to 4, so a repunit in ternary. Furthermore, 121 is the only square of the form $$1 + p + p^2 + p^3 + p^4$$, where p is prime (3, in this case).
 * the sum of three consecutive prime numbers (37 + 41 + 43).
 * As $$5! + 1 = 121$$, it provides a solution to Brocard's problem. There are only two other squares known to be of the form $$n! + 1$$. Another example of 121 being one of the few numbers supporting a conjecture is that Fermat conjectured that 4 and 121 are the only perfect squares of the form $$x^{3}-4$$ (with $x$ being 2 and 5, respectively).
 * It is also a star number, a centered tetrahedral number, and a centered octagonal number.
 * In decimal, it is a Smith number since its digits add up to the same value as its factorization (which uses the same digits) and as a consequence of that it is a Friedman number ($$11^2$$). But it cannot be expressed as the sum of any other number plus that number's digits, making 121 a self number.

In other fields
121 is also:
 * The electricity emergency telephone number in Egypt
 * The number for voicemail for mobile phones on the Vodafone network
 * The undiscovered chemical element unbiunium has the atomic number 121
 * The official end score for cribbage
 * The pennant number of RTS Moskva, the Russian Navy’s Black Sea flagship, which was damaged beyond repair on April 13, 2022.

title: 121 (number)


text:
Talk:Argument from inconsistent revelations

Votes for Deletion

 * Argument_against_any_specific_God - Although I am an atheist myself, an article with this title cannot exist without being POV. Atheist rant. Mrdice 09:21, 2004 Feb 23 (UTC)
 * POV Atheist rant. Delete. Optim 09:35, 23 Feb 2004 (UTC)
 * Delete rant -- Graham :) 11:14, 23 Feb 2004 (UTC)
 * Delete. Take any relevant information and paste on Atheism or Agnosticism if possible.  Oberiko 12:50, 23 Feb 2004 (UTC)
 * Delete. Original research. Anthony DiPierro 14:28, 23 Feb 2004 (UTC)
 * Delete. Original...er....research. DJ Clayworth 14:40, 23 Feb 2004 (UTC)
 * Delete. POV rant.  Seth Mahoney 19:41, 23 Feb 2004 (UTC)
 * Move to Argument from inconsistent revelations, fix, and keep. We have pages for each of several Arguments for the existence of God, and a page on Arguments against the existence of God as well.  This article needs help to be sure, but it contains the gist of that argument. -- Smerdis of Tlön 20:06, 23 Feb 2004 (UTC)
 * Smerdis/Ihcoyc is right -- keep. It definitely needs an overhaul, but I can envision this article outlaying the argument (and responses of course) in NPOV fashion. Jwrosenzweig 20:39, 23 Feb 2004 (UTC)
 * Delete, Smerdis is wrong, Argument from inconsistent revelations is original "research" (more like poorly thought out musings), and, incidentilly, is pure bunk ;) If you would like to change the policy on original research go talk to Jimbo. Sam Spade 20:56, 23 Feb 2004 (UTC)
 * Keep. I don't see how this counts as original research. There is no actual research in it. Secretlondon 22:48, Feb 23, 2004 (UTC)
 * Fix, but if not fixed, delete. Definitely not NPOV (and i say this as a militant athiest). Morwen 23:00, Feb 23, 2004 (UTC)
 * Delete. Everyking 23:50, 23 Feb 2004 (UTC)
 * Delete but if not fixed AY 05:25, 24 Feb 2004 (UTC)
 * Keep. Can be made suitably NPOV. WP has Anselm's ontological argument, it can have the counter-arguments as well. Wile E. Heresiarch 14:39, 24 Feb 2004 (UTC)
 * This has nothing to do with Anselm's argument, which is appropriate for inclusion in WP for historical reasons. Were it established that this argument was a rebuttal to Anselm's by some prominent philosopher or something similar, it might be appropriate for inclusion with sufficient NPOVing, but this isn't the case. Seth Mahoney 18:23, 24 Feb 2004 (UTC)
 * You're absolutely right, it's irrelevant for me to bring Anselm into this. I still believe the page can be made suitably NPOV. Wile E. Heresiarch 02:56, 25 Feb 2004 (UTC)
 * Its interesting that you're supporting keeping an article here that definately qualifies as original research (as written, it contains no references as to who it originally came from, where it has been used, etc.), yet you support deleting Fractional probability on the grounds that it is original research. Admittedly, this article makes more sense than Fractional probability, but it is still original research.  -Seth Mahoney 19:01, 25 Feb 2004 (UTC)
 * The tittle "Argument against any specific God" is not in itself POV! 129.240.69.213 05:56, 26 Feb 2004 (UTC)
 * The question is always the same: is this a report or an essay? A report on how philosophers have treated and are treating such an argument is Wikipedia material. An essay on whether the subject itself is right or wrong is not encyclopedia material. Quite simple. Wetman 20:25, 26 Feb 2004 (UTC)
 * I am not sure I understand this objection well enough to be able to improve the article to respond to it. I have attempted to make this article roughly conformable to the other Arguments for the existence of God and Arguments against the existence of God, setting out the argument and potential responses or objections to it.  I don't see how this one advocates a particular position more or less than the other, comparable pages do.  Smerdis of Tlön 14:30, 27 Feb 2004 (UTC)
 * Keep with cleaning-up. It's a valid philosophical arguement and historical names have been added to back it up.TimothyPilgrim 14:26, Mar 22, 2004 (UTC)

Discussing the merits of the article itself: is the stuff about prayer really relevant? It seems to be starting a whole new argument, one that doesn't actually rely on conflicting revelations. For example two Christians who both believe exactly the same thing about God could be praying for contradictory things (such as to get the same job). I'm not saying the prayer question isn't an issue, but that it's a different issue. DJ Clayworth 15:20, 27 Feb 2004 (UTC)

 * That was in the article as originally written. There are in fact two "points" to be made by this line of reasoning.  The first, that inconsistent revelations, where you damn yourself according to one faith by accepting another, alter the payoff matrix of Pascal's Wager.  The second line of reasoning states that if an all powerful God chooses to reveal a plan of salvation, the revelation should be so unmistakable and convincing that all who consider it know it is from God.  The existence of disagreements about which faith to follow, the possibility of contradictory prayers, and similar contradictions in practice suggest that no all powerful god has revealed a true religion to humans.  -- Smerdis of Tlön 05:30, 2 Mar 2004 (UTC)

Added an additional point to the argument. I was torn between the bold statement of the argument (which I went with) and a weaselly 'some critics of the argument say'. If anyone can improve on my phraseology please do so. DJ Clayworth 15:27, 27 Feb 2004 (UTC)

In an effort to keep this page extant because I think it is a valuable part of philosophy, I've revised a lot of grammar, punctuation, and cleared up some points. However, I didn't touch the part on prayer and the paragraphs following it. I agree with others in that the article does not benefit from the inclusion of the problem of prayer, that is suitable for a different article, and the following text does not make sense to me. I can't tell what's being referred to, and I didn't want to revise it for fear of changing the intended meaning. Simply put, do the final paragraphs deal with the title article, or the opposition to it? Would the original author please clarify this? Comments? TimothyPilgrim 16:10, Mar 22, 2004 (UTC) Delete the math section atleast. An equation can't determine salvation or the true path to God. That makes no sense. —Preceding unsigned comment added by 68.58.250.209 (talk) 18:58, 17 March 2011 (UTC)

Rebuttal is logically inconsistent
I removed the section below because it implies that if someone claiming to be a witness to revelation is found to have told a lie, or made an inaccurate prediction then he did not witness the revelation, and that if he is found to have made an accurate prediction then he did witness revelation. I predicted the colts to win the 2007 super bowl, but that doesnt mean my statements about God are true. The fallacy here is an appeal to authority, I believe.

Removed paragraph: Believers have a number of stratagems to counter this argument. It assumes, for example, that none of them make verifiable predictions about what can be found in history or science. The presence of a testable proposition in a revelation may provide a way to assess the credentials of the prophet who claims to speak for a deity; an error about an inter-subjectively demonstrable fact casts doubt on the remaining propositions that cannot be verified. —The preceding unsigned comment was added by Tritium6 (talk • contribs) 20:05, 17 April 2007 (UTC).

Logical Problems
I see two logical problems with this argument. One is that it seems to be saying all revelations can't be true and then implies because of that none should be believed to be true. However isn't this the fallacy of the Excluded middle since its certainly possible that one is true and all the rest false?

Secondly, you could apply the probability argument to atheism or agnosticism as well, since you could argue choosing them is also probably in incorrect choice. I can't be the first to realize these things so is there any sources that make these cases that could be included in the article as it needs a balance in my opinion?67.161.205.170 (talk) 09:06, 17 April 2009 (UTC)
 * Regarding the first, it can for example be seen as a counter-argument to Pascal's wager. Secondly, agnosticism is probably the correct choice since it requires the least assumptions given what evidence we have.Ht686rg90 (talk) 10:45, 17 April 2009 (UTC)
 * I think this argument is intended to point out that arguments and revelations on their own are not sufficient to prove that any given God exists without corroborative evidence to back them up- if different people have "revelations" for different Gods, then revelations aren't enough to prove that one God exists. As for the excluded middle, what this argument is saying is that if a given argument for the existence for God works for multiple religions, while it may be possible that one possiblity is right and the others wrong, it could also mean that the argument is flawed. This argument is more of a universal counterargument to existing arguments for God's existence. For the second point- atheism and agnosticism rely on different arguments, and all of these point to just atheism- arguments against the existence of God argue against the existence of all Gods equally and point to none of them existing- so inconsistent revelations isn't a counterargument which applies to them.

But, as this is supposed to be about improving the article- I don't know of any major sources which have voiced these objections. I'm an atheist, and while I reckon a counterarguments section would be good, I'm yet to find any major figures who've specifically attacked this argument. When I next get a chance, I might have a look (but that could be a while). In the meantime, you could probably Google the argument and search through the first few pages- you'd be surprised what it can turn up.122.106.160.244 (talk) 12:45, 18 April 2009 (UTC)

 * Yes, it would be great, 122, if such sources could be found, but this is September and all the above words were written last April. What does this say about the argument?  It looks to me as if 67 above is correct.  This isn't really an argument against the existence of God, because God can be taken out of the picture altogether.  Atheism, while not a "religion" per se, still requires "faith" due to the solid gold fact that it is just as unlikely that "hard evidence" can be found that God does not exist as it is that such evidence can be found that God does exist.  So from the  probability point of view as shown mathematically in this article, the chance of an atheist being correct and that all of those "belief in God" faiths are incorrect is essentially 1/(n+1).  The "1" in "n+1" represents the "faith" required by atheism, the faith that there is no God.

"This equation, by the way, also represents the chances of all the 'belief in God' faiths as well, because from the standpoint of probability, all those believers in God might be wrong, and atheists could be correct. So basically we are back to 1/n, where n is equal to all the inconsistent faiths including atheism. Just so it's not as confusing, we can say that d = n+1; this yields an atheism-inclusive probability of 1/d that any one faith, including atheism, is correct."

 * Now, if this "argument from inconsistent revelations" is, in fact, "an argument against the existence of God", and atheists recognize it as such, then either atheists are right about this OR... atheists are wrong and the argument is faulty and incorrect. An atheist must ask her/himself why is it logical not to include atheism in this probability argument?  It is not logical because there is always the >99% probability that at least one of those "belief in God" faiths is correct.  So at one and the same time, if this argument from inconsistent revelations were correct, then an atheist has both a 50-50, 1/2 probability of being correct that God doesn't exist and a >99% chance of going to the hell of one of those "hundreds of religions in existence".  Only one of these conditions can be correct, i.e., an atheist either has a 1/2 chance of going to hell or a d/1 chance of going to hell.  Thus, the argument from inconsistent revelations appears to be defeated.

 * So rather than being "an argument against the existence of God", this argument from inconsistent revelations, which, if correct, must include the absence of revelation, as in atheism (for zero revelations is inconsistent with all the other revelations), this is an argument that falls into the category of "faulty logic". This may explain why no reliable sources can be found.  There may be a >99% probability that notable people who look at this article think it's a waste of their time to bother with it.
 * —  .`^) Paine Ellsworthdiss`cuss (^`.   05:09, 9 September 2009 (UTC)
 * Either there is a god (religion of all types) or there isn't (atheism), therefore an atheist has a 1 in 2 chance of being right. However, if there is a god there is then the question of WHICH one. Therefore any believer has a 1 in 2n chance of being right, where n = the number of contradictory gods and the coefficient 2 represents the halving of the total from separating out atheism. That is, if there are 50 different possible gods a believer has only a 1 percent chance of picking correctly. Khajidha (talk) 17:06, 14 January 2010 (UTC)

I'm REALLY late to this party (the original comment starting this discussion was back in 2007), but I just wanted to point out that whether the arguments themselves are flawed, the purpose of the article is simply to document the argument and its basis. I certainly agree that there are logical fallacies in these arguments, and further that logic and faith are frequently mutually exclusive. But regardless of the validity of the argument or the logical sets that are incorrectly included or excluded, I think this article does a good job of explaining the argument as a whole, and how it is often applied to the topic of atheism. Keep in mind as well that it is one of several major arguments against the existence of God, and those other arguments address the issue from other standpoints. — KieferSkunk (talk) — 19:46, 16 May 2011 (UTC)

"Either there is a god (religion of all types) or there isn't (atheism), therefore an atheist has a 1 in 2 chance of being right." Wrong. Hinduism, Shintoism and as far as I know all folk religions are polytheist so either there are gods, there is a god or there are no gods at all. To say that there are 2 possibilities so one is as likely as the other might be sound mathematically but the mechanics of theology or philosophy are different. Say I play along with the mathematical approach; as I've demonstrated there are 3 possiblities and not 2 so an atheist has a 1/3 chance of being right...and therefore a 2/3 chance of being wrong. "That is, if there are 50 different possible gods a believer has only a 1 percent chance of picking correctly." Dead wrong again and this time mathematically unsound. There are far more gods than religions; as I've pointed out already there are polytheistic faiths; take Shintoism where there is a god in everything; if Shintoism is false then there are a limited number of gods and since Amaterasu is a goddess in Shintoism only then if Shintoism is false then Amaterasu and all of the other deities exclusive to Shintoism don't exist. In other words there isn't an equal chance of God, Amaterasu and Fujin existing. If Shintoism is false then neither Amaterasu nor Fujin exist as they are only gods in the Shinto religion while God is unaffected as he is the deity of the Abrahamic faiths. 86.42.121.148 (talk) 22:37, 15 June 2014 (UTC)

Errors in layout of article
Could someone please tidy up the first paragraph, as the first word is above the entire paragraph. Thankyou. —Preceding unsigned comment added by 78.33.99.227 (talk) 08:37, 7 September 2010 (UTC)

hokus systematic description section
I think the "systematic description" section should be removed. It doesn't cite any sources, it's mathematically dodgy, and it has only given fruit to silly disputes above ("logical problems" thread).

It's dodgy in light of modern probability theory because it inappropriately gives equal prior plausibility to every model in any arbitrary set (say: catholicism, hard atheism, anglicism, islam, flying spaghetti monster) thereby leading to nonsense conclusions (e.g. hard-atheism is precisely three times less probable to be true than the abrahamic deity).

The fact is, it's a complete misinterpretation. The argument is not supposed to be direct support for hard atheism. Instead it is supposed to refute a large class of arguments supporting religion. By nullify all those, this argument constitutes indirect support for soft-atheism (and agnosticism). No more, no less.

(Its consequence for the debate is that it purports to reduce the theist to only using arguments that do not favour her own particular beliefs anyway - except ones based on any rare points where her religion is obviously unique from the set of all conceivable beliefs.) Cesiumfrog (talk) 09:14, 10 June 2011 (UTC)

 * I agree that trying to explain this argument in mathematical terms both dodges and dilutes the real issue being argued. But there is some merit to doing so - it may be difficult to get the general idea of the argument without seeing it in some reasonably concrete terms.  I don't think I know enough about this topic to know if the formula, which basically says "They can't ALL be right", is even close to the real argument or not, but that's always been my impression of it: Since each religion was created by its people and is purely a matter of belief, each religion believes it is the CORRECT one, and someone from the outside with no other evidence would have to look at all of the available religions and decide which one was correct, if any, it's logical to conclude that in fact NONE of them are likely to be correct (there's an increasingly small chance of any one of them being the correct one, as the number of faiths increases).  That sort of logical argument is possible to express mathematically, but of course reducing it to that eliminates a lot of the nuances that would likely affect one's choice of belief (whether to believe, what to believe, etc.). — KieferSkunk (talk) — 20:31, 10 June 2011 (UTC)

 * I agree that the "systematic description" should be either deleted or seriously edited by someone with a solid handle on the argument. The argument presented in the "systematic description" is not systematic, and it begins with the phrase "in mathematical terms" and the proceeds to use non-mathematical descriptions in its non-systematic exposition of the argument. If the consensus is that the description section should be deleted, then I support deleting the entire article. If the article can be substantially rewritten it may be worth saving since there is indeed some professional literature on this problem. 76.119.208.81 (talk) 22:23, 26 June 2011 (UTC)

Systematic description section- edits made
I have removed "in mathematical terms" from the opening to this section because:
 * Most of the terms used in the section are not mathematical.
 * It is not normal protocol to identify what language one is using at the outset of each use of that language.

I have rephrased "faiths" to "interpretations of that God" to improve the neutrality of the article and because any systematic presentation of an argument should use as few different major terms as possible; it is easier to see the flow of the argument if the word "god" appears in each step.

I have removed its statement that "each of these faiths has a corresponding Hell" because:
 * This is a factually incorrect statement.
 * This is a non-neutral interpretation of contested concepts in the major religions probably intended to be the referents of this section.
 * Hell does not appear elsewhere in the argument and so this premise seems to have no bearing on the conclusion or on any other premise.

I have changed "in practice" to "since" because "in practice" means "within the context of real-world activity," not "in the real world" and "since" is the more succinct statement of this intended rule. I have also removed language about picking religions "at random" and clarified the sentence to more clearly state that the point of the argument is that, a priori, religions appear on their face to contradict, but not that the choice between religions is a pure crapshoot, but rather that the argument as currently formulated proceeds from a priori terms without considering any other claims.

I also strongly recommend that someone with access to professional sources add how this argument is I believe intended, as a negative reflection on God's goodness to allow religions about him to proliferate so rapidly and so inconsistently. 76.119.208.81 (talk) 22:40, 26 June 2011 (UTC)

 * I'm sure you have made the section better than it was before. Unfortunately, it still looks like original research. Can you provide any reputable external reference for presenting the argument in that way?

 * Just to pick one example of why the improved argument is still wrong, why would anyone assign equal prior probability to every deity hypothesis? (For example, if a more reasoned person were to assign each a numerical probability, mightn't they instead base it on say proportion of proponents?)

 * Or to be technical, the completion of your argument should include infinitely many hypotheses (since the fact that some have no believers nor even proposers needn't disqualify them from consideration) but a uniform distribution over an infinite set is an invalid prior.

 * Casting the argument in your "systematic" form tends to lead the reader to conclude that the Abrahamic deity exists, more likely than not, since some variation is included in such a large proportion of the "hundreds of religions in existence" presently. This is the opposite of the desired conclusion. Instead, the point is supposed to be: "religionist A's arguments are invalid, because religionist B uses the same arguments for an incompatible hypothesis. This does not say anything about whether deities exist, but does prove that the religionists can tell us nothing either way." The problem is that it sounds like you are trying to use it to say something about whether deities exist, and not just about whether religionists are qualified to inform us of the personal characteristics if they do.

 * By the way, could you clarify what you meant in the last sentence (mentioning "negative reflection on God's goodness" and proliferation?) Cesiumfrog (talk) 00:15, 27 June 2011 (UTC)

 * I think he means that part of the argument is essentially, "If a God really does exist, why are there so many different religions that have different gods, different modes of belief, etc.?" An interesting question, but I'm not sure if it falls under this argument or a different one. — KieferSkunk (talk) — 20:15, 27 June 2011 (UTC)

 * I concur with the above and suggest that Diderot, Denis (1875-77) [1746]. J. Assézar. ed (in French). Pensées philosophiques, LIX, Volume 1. pp. 167 be used as a citation for this argument, wherein Diderot makes the argument to which this article alludes. 98.249.14.80 (talk) 20:18, 2 July 2011 (UTC)

Deism vs. theism in this argument
Where mentioned, theistic texts that render some version of theism exclusive over all others should cite to passages within those scriptures or major theological works that are apropos to such a point. Similarly, the proposed immunity of deism to this argument should either discuss, link to, or cite a discussion of the problem on this point that deism, henotheism, and polytheism are incompatible with each other but are all compatible with the argument from inconsistent revelation. Thomas Paine discusses but dismisses this point in two or three very brief passages in I think a formally invalid way, and there is some interesting literature in Greek philosophy, especially the latter works of Plato on Socrates, on this point. In short, more citations please! 76.118.32.67 (talk) 12:34, 9 July 2011 (UTC)

Logic error in mathematical description
There is a glaring logic error in the mathematical description, in that it seems to assume that the correct descriptor must come from the set of descriptors that men know about. But it is entirely possible that 1) the exists a 'God' but 2) no human religion provides a correct descriptor of this entity, and so the presumption of there being a God does not require that this God come from the 1/n set of beliefs held by people, but from the 1/y set of all possible descriptors of a God. DeistCosmos (talk) 17:13, 8 December 2012 (UTC)

Dubious
Yes the Torah does say not to worship any other gods but as far as I know all of the other gods it names it says are false gods. It certainly says that about Baal and Moloch anyway. 86.42.121.148 (talk) 01:22, 16 June 2014 (UTC)

Same arguments? Confusion....
Is the argument from inconsistent revelations the same as the argument from religious confusion, or the problem of religious diversity? Τζερόνυμο (talk) 19:08, 18 May 2018 (UTC)
 * They seem like alternate names for the same argument, yes.Mcc1789 (talk) 04:24, 10 June 2019 (UTC)
 * I think renaming the article could be a good idea. If I plug "argument from inconsistent revelations" in Google scholar, I don't find anything. If I plug in "problem of religious diversity", there is at least a book and several articles (not so with "argument from religious confusion". I think we could help solving the dearth of sources in this article by changing the title and mining the sources about the "problem of religious diversity". MonsieurD (talk) 15:24, 17 July 2020 (UTC)
 * There is also alot of material about the same concept here on Wikipedia. Another reliable and reputable source about a similar problem can be found here. MonsieurD (talk) 19:20, 17 September 2020 (UTC)
 * Hmm.. the article says that this is an argument against the existence of god without a source. I don't see how this could end up being about the existence of god. It would be an argument against a wrong god or exclusivistic religion, but not really argument against the existence of god. The plurality of cultures is not an argument for the nonexistence of cultures and the plurality of religions is not an argument against the existence of religions either.Ramos1990 (talk) 23:09, 17 September 2020 (UTC)
 * Hi, I've changed the lede to make it say that it's an argument about the possibility to know if any single religion is the right one. I think that's closer to the idea. Tell me what you think.MonsieurD (talk) 13:27, 18 September 2020 (UTC)
 * It makes more sense, and is more in line with one of the sources you mentioned. But I don't really see how this articles could be related to atheism in the infobox. Furthermore, it does not really seem to be a problem for most religions except the handful of exlcusivistic religion - which is already dealt with by them already (e.g. Christainity, missionaries, apologetics and the Tower of Babel) and does not appear to be a legitimate issue. The fact that very few sources are found for this means that it is not really a stand alone topic in academic literature. At least this article may need to include religious solutions to pluralism that already exist.Ramos1990 (talk) 03:27, 19 September 2020 (UTC)

Deleted content
Hi folks,

has deleted a large part of [| this edit] for the following reason : "Source does not treat the argument at all since this is an argument used against the existence of god - not an argument for pluralism or against pluralism. for pluralism stuff it belongs in the pluralism page".

I think he's wrong. The content I added has nothing to do with being for or against pluralism, but tries to derive the consequences of pluralism, namely, what is a believer to do with the fact that other people from other religions have just as good reasons to believe in their religions than him?

To me, this is exactly the argument from inconsistent revelation. By adding this content, I was trying to anchor this Wikipedia article into an actual debate within philosophy, which is a problem with this article. If you plug "Argument from inconsistent revelations" in Google Scholar, you get nothing by that name. It's a a good, intuitive name, but it doesn't correspond to how people refer to the problem/argument in the litterature.

This was already discussed [| here].

So, what should we do with this article? Is the content I added relevant or not? Should we rename the article? If we don't we will have to deal with the fact that we can't find sources for this article which correspond to its title on Wikipedia--MonsieurD (talk) 12:44, 2 April 2022 (UTC)
 * Your edit was literally about religious pluralism and would be better fit in that page as that was the scope of the source itself - literally . It does not treat the argument about inconsistent revelations as an argument against the existence of God, which is what this article specifically tries to address. See the sources on the article. Richard Dawkins seems to be the only contemporary source for it and he is not a reliable source on religion.

 * I think that this article has no real content since there really are no sources on it specifically, like you mentioned, and should be either deleted or merged with the religious pluralism. At least then it would be anchored to the literature and could be a section on the religious pluralism page. What do you think?Ramos1990 (talk) 17:46, 2 April 2022 (UTC)


text:
Peep show

A peep show or peepshow is a presentation of a live sex show or pornographic film which is viewed through a viewing slot.

Several historical media provided voyeuristic entertainment through hidden erotic imagery. Before the development of the cinema in 1895, motion pictures were presented in peep boxes, such as the kinetoscope and the mutoscope. These remained relatively popular for erotic and pornographic films, such as What the Butler Saw.

In contemporary use, a peep show is a piecewise presentation of pornographic films or a live sex show which is viewed through a viewing slot, which shuts after the time paid for has expired. The viewing slots can be operated by a money box device, or paid for at a counter.

Pornographic peep shows became popular in the 1970s as part of the developing pornography industry. Until home video became widespread, peep shows made up a major part of the way in which video pornography was accessed. In 1986 a US Presidential report into pornography said that peep shows were making significant earnings which were often undeclared or untaxed, and in some US locations peep shows were subsequently suppressed.

For live peep shows, booths can surround a stage upon which usually a female performer performs a striptease and sexually explicit poses. In Barcelona female performers at times also perform sexual intercourse with male performers on stage. In some cases, booths include paper towel dispensers for customers who engage in masturbation. A customer and performer can mutually agree on a fee for a "private dance", which can take place in a peep show booth with a clear window and seating space for only one spectator.

California

Research on peep show establishments in California examined the hypothesis that neighborhoods surrounding sex businesses such as peep show establishments and X-rated movie stores have higher rates of crime. The researchers compared 911 calls in peep show and control neighborhoods in San Diego. Although peep show neighborhoods had approximately 16 percent more calls, the researchers concluded that the difference was not statistically significant. Other researchers reanalyzed the data and concluded that the difference was significant.

Regal Show World
Regal Show World was an adult entertainment business on lower Market Street in San Francisco, California. The company's slogan was "Where you are king". The business had a peep show and an adult video arcade. The peep show had performers working in an enclosed round room with viewing booths surrounding it and sometimes had a "double in the bubble" show in which two performers worked simultaneously. During the winter of 1997 to 1998, the business had thirty-five performers. At this time, over 80% of performers there attempted to unionize and "signed union authorization cards for representation by the Service Employees International Union (SEIU), Local 790".

The business was owned by Bijou Group, Inc., a privately held company in San Francisco that was founded in 1990. Bijou Group owned similar businesses in San Francisco such as New Century Theater, Market Street Cinema, and the Campus Theater. Bijou Group, Inc. filed for Chapter 11 bankruptcy reorganization c. 1994. In November 1998, management of Regal Show World announced that the peep show would be closed on November 30 of that month due to "economic reasons". At the time, some performers in the industry stated that closure of the peep show was done as a retaliative measure against attempts for performers to unionize. The company declared bankruptcy after performers made a second attempt at unionization, whereby the performers "signed cards calling for a union election", and the theater was closed.

Lusty Lady
The former Lusty Lady peep show in San Francisco, California, entered the news in 1997, when it became the first U.S. sex business to be unionized. In 2003 it was bought by the employees and became a worker cooperative. It closed due to increased rent in 2013.

Nevada
In Las Vegas in the early 1990s, city authorities began to move peep shows and other sexually oriented businesses away from the city centre. The last peep show in Las Vegas closed in 2019.

New York
Times Square in New York was famous for its peep shows up until the 1990s. Female performers would stand undressed on a small semicircular or circular stage surrounded by individual booths. By inserting a token into a machine, a patron could open the opaque partitions separating them from the stage. He could then choose a performer and, for the payment of a small tip, summon her and fondle her sexually or ask her to perform sex acts on him. After taking office as mayor in 1994, Rudy Giuliani waged a legal battle to shut down the Times Square peep shows. Following a major legal decision in Giuliani's favor in 1998, the peep shows mostly closed.

Washington
The former Lusty Lady peep show in Seattle, Washington, was, unlike its namesake in San Francisco, not unionized. Similar to the Lusty Lady in San Francisco, however, the Seattle peep show was deemed an iconic local landmark by the time it finally closed in 2010. Its closing was attributed to being unable to compete with pornography on the Internet.

In popular culture
American singer Madonna plays a peep show dancer in the video for her song Open Your Heart

title: Peep show