Lemniscate constant

In mathematics, the lemniscate constant $ϖ$ is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of $π$ for the circle.^[1] Equivalently, the perimeter of the lemniscate $(x^{2}+y^{2})^{2}=x^{2}-y^{2}$ is $2 ϖ$ . The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755.^[2] It also appears in evaluation of the gamma and beta function at certain rational values. The symbol $ϖ$ is a cursive variant of $π$ ; see Pi § Variant pi.

Sometimes the quantities $2 ϖ$ or $ϖ/2$ are referred to as the lemniscate constant.^[3]^[4]

As of 2024 over 1.2 trillion digits of this constant have been calculated.^[5]

History

Gauss's constant, denoted by G, is equal to $ϖ / π \approx 0.8346268$ ^[6] and named after Carl Friedrich Gauss, who calculated it via the arithmetic–geometric mean as $1/M{\bigl (}1,{\sqrt {2}}{\bigr )}$ .^[7] By 1799, Gauss had two proofs of the theorem that $M{\bigl (}1,{\sqrt {2}}{\bigr )}=\pi /\varpi$ where $\varpi$ is the lemniscate constant.^[8]

John Todd named two more lemniscate constants, the first lemniscate constant $A = ϖ /2 \approx 1.3110287771$ and the second lemniscate constant $B = π /(2 ϖ) \approx 0.5990701173$ .^[9]^[10]^[11]

The lemniscate constant $\varpi$ and Todd's first lemniscate constant $A$ were proven transcendental by Carl Ludwig Siegel in 1932 and later by Theodor Schneider in 1937 and Todd's second lemniscate constant $B$ and Gauss's constant $G$ were proven transcendental by Theodor Schneider in 1941.^[9]^[12]^[13] In 1975, Gregory Chudnovsky proved that the set $\{\pi ,\varpi \}$ is algebraically independent over $\mathbb {Q}$ , which implies that $A$ and $B$ are algebraically independent as well.^[14]^[15] But the set ${\bigl \{}\pi ,M{\bigl (}1,1/{\sqrt {2}}{\bigr )},M'{\bigl (}1,1/{\sqrt {2}}{\bigr )}{\bigr \}}$ (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over $\mathbb {Q}$ .^[16] In 1996, Yuri Nesterenko proved that the set $\{\pi ,\varpi ,e^{\pi }\}$ is algebraically independent over $\mathbb {Q}$ .^[17]

Forms

Usually, $\varpi$ is defined by the first equality below, but it has many equivalent forms:^[18]

${\begin{aligned}\varpi &=2\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}={\sqrt {2}}\int _{0}^{\infty }{\frac {\mathrm {d} t}{\sqrt {1+t^{4}}}}=\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {t-t^{3}}}}=\int _{1}^{\infty }{\frac {\mathrm {d} t}{\sqrt {t^{3}-t}}}\\[6mu]&=4\int _{0}^{\infty }{\Bigl (}{\sqrt[{4}]{1+t^{4}}}-t{\Bigr )}\,\mathrm {d} t=2{\sqrt {2}}\int _{0}^{1}{\sqrt[{4}]{1-t^{4}}}\mathop {\mathrm {d} t} =3\int _{0}^{1}{\sqrt {1-t^{4}}}\,\mathrm {d} t\\[2mu]&=2K(i)={\tfrac {1}{2}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )}={\tfrac {1}{2{\sqrt {2}}}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{4}}{\bigr )}={\frac {\Gamma (1/4)^{2}}{2{\sqrt {2\pi }}}}={\frac {2-{\sqrt {2}}}{4}}{\frac {\zeta (3/4)^{2}}{\zeta (1/4)^{2}}}\\[5mu]&=2.62205\;75542\;92119\;81046\;48395\;89891\;11941\ldots ,\end{aligned}}$

where $K$ is the complete elliptic integral of the first kind with modulus $k$ , $Β$ is the beta function, $Γ$ is the gamma function and $ζ$ is the Riemann zeta function.

The lemniscate constant can also be computed by the arithmetic–geometric mean $M$ ,

$\varpi ={\frac {\pi }{M{\bigl (}1,{\sqrt {2}}{\bigr )}}}.$

Gauss's constant is typically defined as the reciprocal of the arithmetic–geometric mean of 1 and the square root of 2, after his calculation of $M{\bigl (}1,{\sqrt {2}}{\bigr )}$ published in 1800:^[19] $G={\frac {1}{M{\bigl (}1,{\sqrt {2}}{\bigr )}}}$ John Todd's lemniscate constants may be given in terms of the beta function B: ${\begin{aligned}A&={\frac {\varpi }{2}}={\tfrac {1}{4}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )},\\[3mu]B&={\frac {\pi }{2\varpi }}={\tfrac {1}{4}}\mathrm {B} {\bigl (}{\tfrac {1}{2}},{\tfrac {3}{4}}{\bigr )}.\end{aligned}}$

As a special value of L-functions

$\beta '(0)=\log {\frac {\varpi }{\sqrt {\pi }}}$

which is analogous to

$\zeta '(0)=\log {\frac {1}{\sqrt {2\pi }}}$

where $\beta$ is the Dirichlet beta function and $\zeta$ is the Riemann zeta function.^[20]

Analogously to the Leibniz formula for π, $\beta (1)=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n}}={\frac {\pi }{4}},$ we have^[21]^[22]^[23]^[24]^[25] $L(E,1)=\sum _{n=1}^{\infty }{\frac {\nu (n)}{n}}={\frac {\varpi }{4}}$ where $L$ is the L-function of the elliptic curve $E:\,y^{2}=x^{3}-x$ over $\mathbb {Q}$ ; this means that $\nu$ is the multiplicative function given by $\nu (p^{n})={\begin{cases}p-{\mathcal {N}}_{p},&p\in \mathbb {P} ,\,n=1\\[5mu]0,&p=2,\,n\geq 2\\[5mu]\nu (p)\nu (p^{n-1})-p\nu (p^{n-2}),&p\in \mathbb {P} \setminus \{2\},\,n\geq 2\end{cases}}$ where ${\mathcal {N}}_{p}$ is the number of solutions of the congruence $a^{3}-a\equiv b^{2}\,(\operatorname {mod} p),\quad p\in \mathbb {P}$ in variables $a,b$ that are non-negative integers ( $\mathbb {P}$ is the set of all primes). Equivalently, $\nu$ is given by $F(\tau )=\eta (4\tau )^{2}\eta (8\tau )^{2}=\sum _{n=1}^{\infty }\nu (n)q^{n},\quad q=e^{2\pi i\tau }$ where $\tau \in \mathbb {C}$ such that $\operatorname {\Im } \tau >0$ and $\eta$ is the eta function.^[26]^[27]^[28] The above result can be equivalently written as $\sum _{n=1}^{\infty }{\frac {\nu (n)}{n}}e^{-2\pi n/{\sqrt {32}}}={\frac {\varpi }{8}}$ (the number $32$ is the conductor of $E$ ) and also tells us that the BSD conjecture is true for the above $E$ .^[29] The first few values of $\nu$ are given by the following table; if $1\leq n\leq 113$ such that $n$ doesn't appear in the table, then $\nu (n)=0$ : ${\begin{array}{|c|c|c|c|}\hline n&\nu (n)&n&\nu (n)\\\hline 1&1&53&14\\\hline 5&-2&61&-10\\\hline 9&-3&65&-12\\\hline 13&6&73&-6\\\hline 17&2&81&9\\\hline 25&-1&85&-4\\\hline 29&-10&89&10\\\hline 37&-2&97&18\\\hline 41&10&101&-2\\\hline 45&6&109&6\\\hline 49&-7&113&-14\\\hline \end{array}}$

As a special value of other functions

Let $\Delta$ be the minimal weight level $1$ new form. Then^[30] $\Delta (i)={\frac {1}{64}}\left({\frac {\varpi }{\pi }}\right)^{12}.$ The $q$ -coefficient of $\Delta$ is the Ramanujan tau function.

Series

Viète's formula for $π$ can be written:

${\frac {2}{\pi }}={\sqrt {\frac {1}{2}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}}}\cdots$

An analogous formula for $ϖ$ is:^[31]

${\frac {2}{\varpi }}={\sqrt {\frac {1}{2}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\bigg /}\!{\sqrt {\frac {1}{2}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\Bigg /}\!{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\bigg /}\!{\sqrt {\frac {1}{2}}}}}}}\cdots$

The Wallis product for $π$ is:

${\frac {\pi }{2}}=\prod _{n=1}^{\infty }\left(1+{\frac {1}{n}}\right)^{(-1)^{n+1}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)={\biggl (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\biggr )}{\biggl (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\biggr )}{\biggl (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\biggr )}\cdots$

An analogous formula for $ϖ$ is:^[32]

${\frac {\varpi }{2}}=\prod _{n=1}^{\infty }\left(1+{\frac {1}{2n}}\right)^{(-1)^{n+1}}=\prod _{n=1}^{\infty }\left({\frac {4n-1}{4n-2}}\cdot {\frac {4n}{4n+1}}\right)={\biggl (}{\frac {3}{2}}\cdot {\frac {4}{5}}{\biggr )}{\biggl (}{\frac {7}{6}}\cdot {\frac {8}{9}}{\biggr )}{\biggl (}{\frac {11}{10}}\cdot {\frac {12}{13}}{\biggr )}\cdots$

A related result for Gauss's constant ( $G=\varpi /\pi$ ) is:^[33]

${\frac {\varpi }{\pi }}=\prod _{n=1}^{\infty }\left({\frac {4n-1}{4n}}\cdot {\frac {4n+2}{4n+1}}\right)={\biggl (}{\frac {3}{4}}\cdot {\frac {6}{5}}{\biggr )}{\biggl (}{\frac {7}{8}}\cdot {\frac {10}{9}}{\biggr )}{\biggl (}{\frac {11}{12}}\cdot {\frac {14}{13}}{\biggr )}\cdots$

An infinite series discovered by Gauss is:^[34]

${\frac {\varpi }{\pi }}=\sum _{n=0}^{\infty }(-1)^{n}\prod _{k=1}^{n}{\frac {(2k-1)^{2}}{(2k)^{2}}}=1-{\frac {1^{2}}{2^{2}}}+{\frac {1^{2}\cdot 3^{2}}{2^{2}\cdot 4^{2}}}-{\frac {1^{2}\cdot 3^{2}\cdot 5^{2}}{2^{2}\cdot 4^{2}\cdot 6^{2}}}+\cdots$

The Machin formula for $π$ is ${\textstyle {\tfrac {1}{4}}\pi =4\arctan {\tfrac {1}{5}}-\arctan {\tfrac {1}{239}},}$ and several similar formulas for $π$ can be developed using trigonometric angle sum identities, e.g. Euler's formula ${\textstyle {\tfrac {1}{4}}\pi =\arctan {\tfrac {1}{2}}+\arctan {\tfrac {1}{3}}}$ . Analogous formulas can be developed for $ϖ$ , including the following found by Gauss: ${\tfrac {1}{2}}\varpi =2\operatorname {arcsl} {\tfrac {1}{2}}+\operatorname {arcsl} {\tfrac {7}{23}}$ , where $\operatorname {arcsl}$ is the lemniscate arcsine.^[35]

The lemniscate constant can be rapidly computed by the series^[36]^[37]

\varpi =2^{-1/2}\pi {\biggl (}\sum _{n\in \mathbb {Z} }e^{-\pi n^{2}}{\biggr )}^{2}=2^{1/4}\pi e^{-\pi /12}{\biggl (}\sum _{n\in \mathbb {Z} }(-1)^{n}e^{-\pi p_{n}}{\biggr )}^{2}

where $p_{n}={\tfrac {1}{2}}(3n^{2}-n)$ (these are the generalized pentagonal numbers). Also^[38]

\sum _{m,n\in \mathbb {Z} }e^{-2\pi (m^{2}+mn+n^{2})}={\sqrt {1+{\sqrt {3}}}}{\dfrac {\varpi }{12^{1/8}\pi }}.

In a spirit similar to that of the Basel problem,

\sum _{z\in \mathbb {Z} [i]\setminus \{0\}}{\frac {1}{z^{4}}}=G_{4}(i)={\frac {\varpi ^{4}}{15}}

where $\mathbb {Z} [i]$ are the Gaussian integers and $G_{4}$ is the Eisenstein series of weight ⁠ $4$ ⁠ (see Lemniscate elliptic functions § Hurwitz numbers for a more general result).^[39]

A related result is

\sum _{n=1}^{\infty }\sigma _{3}(n)e^{-2\pi n}={\frac {\varpi ^{4}}{80\pi ^{4}}}-{\frac {1}{240}}

where $\sigma _{3}$ is the sum of positive divisors function.^[40]

In 1842, Malmsten found

\beta '(1)=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {\log(2n+1)}{2n+1}}={\frac {\pi }{4}}\left(\gamma +2\log {\frac {\pi }{\varpi {\sqrt {2}}}}\right)

where $\gamma$ is Euler's constant and $\beta (s)$ is the Dirichlet-Beta function.

The lemniscate constant is given by the rapidly converging series

$\varpi =\pi {\sqrt[{4}]{32}}e^{-{\frac {\pi }{3}}}{\biggl (}\sum _{n=-\infty }^{\infty }(-1)^{n}e^{-2n\pi (3n+1)}{\biggr )}^{2}.$

The constant is also given by the infinite product

\varpi =\pi \prod _{m=1}^{\infty }\tanh ^{2}\left({\frac {\pi m}{2}}\right).

Also^[41]

\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{6635520^{n}}}{\frac {(4n)!}{n!^{4}}}={\frac {24}{5^{7/4}}}{\frac {\varpi ^{2}}{\pi ^{2}}}.

Continued fractions

A (generalized) continued fraction for $π$ is ${\frac {\pi }{2}}=1+{\cfrac {1}{1+{\cfrac {1\cdot 2}{1+{\cfrac {2\cdot 3}{1+{\cfrac {3\cdot 4}{1+\ddots }}}}}}}}$ An analogous formula for $ϖ$ is^[10] ${\frac {\varpi }{2}}=1+{\cfrac {1}{2+{\cfrac {2\cdot 3}{2+{\cfrac {4\cdot 5}{2+{\cfrac {6\cdot 7}{2+\ddots }}}}}}}}$

Define Brouncker's continued fraction by^[42] $b(s)=s+{\cfrac {1^{2}}{2s+{\cfrac {3^{2}}{2s+{\cfrac {5^{2}}{2s+\ddots }}}}}},\quad s>0.$ Let $n\geq 0$ except for the first equality where $n\geq 1$ . Then^[43]^[44] ${\begin{aligned}b(4n)&=(4n+1)\prod _{k=1}^{n}{\frac {(4k-1)^{2}}{(4k-3)(4k+1)}}{\frac {\pi }{\varpi ^{2}}}\\b(4n+1)&=(2n+1)\prod _{k=1}^{n}{\frac {(2k)^{2}}{(2k-1)(2k+1)}}{\frac {4}{\pi }}\\b(4n+2)&=(4n+1)\prod _{k=1}^{n}{\frac {(4k-3)(4k+1)}{(4k-1)^{2}}}{\frac {\varpi ^{2}}{\pi }}\\b(4n+3)&=(2n+1)\prod _{k=1}^{n}{\frac {(2k-1)(2k+1)}{(2k)^{2}}}\,\pi .\end{aligned}}$ For example, ${\begin{aligned}b(1)&={\frac {4}{\pi }},&b(2)&={\frac {\varpi ^{2}}{\pi }},&b(3)&=\pi ,&b(4)&={\frac {9\pi }{\varpi ^{2}}}.\end{aligned}}$

In fact, the values of $b(1)$ and $b(2)$ , coupled with the functional equation $b(s+2)={\frac {(s+1)^{2}}{b(s)}},$ determine the values of $b(n)$ for all $n$ .

Simple continued fractions

Simple continued fractions for the lemniscate constant and related constants include^[45]^[46] ${\begin{aligned}\varpi &=[2,1,1,1,1,1,4,1,2,\ldots ],\\[8mu]2\varpi &=[5,4,10,2,1,2,3,29,\ldots ],\\[5mu]{\frac {\varpi }{2}}&=[1,3,4,1,1,1,5,2,\ldots ],\\[2mu]{\frac {\varpi }{\pi }}&=[0,1,5,21,3,4,14,\ldots ].\end{aligned}}$

Integrals

A geometric representation of

\varpi /2

and

\varpi /{\sqrt {2}}

The lemniscate constant $ϖ$ is related to the area under the curve $x^{4}+y^{4}=1$ . Defining $\pi _{n}\mathrel {:=} \mathrm {B} {\bigl (}{\tfrac {1}{n}},{\tfrac {1}{n}}{\bigr )}$ , twice the area in the positive quadrant under the curve $x^{n}+y^{n}=1$ is ${\textstyle 2\int _{0}^{1}{\sqrt[{n}]{1-x^{n}}}\mathop {\mathrm {d} x} ={\tfrac {1}{n}}\pi _{n}.}$ In the quartic case, ${\tfrac {1}{4}}\pi _{4}={\tfrac {1}{\sqrt {2}}}\varpi .$

In 1842, Malmsten discovered that^[47]

$\int _{0}^{1}{\frac {\log(-\log x)}{1+x^{2}}}\,dx={\frac {\pi }{2}}\log {\frac {\pi }{\varpi {\sqrt {2}}}}.$

Furthermore, $\int _{0}^{\infty }{\frac {\tanh x}{x}}e^{-x}\,dx=\log {\frac {\varpi ^{2}}{\pi }}$

and^[48]

$\int _{0}^{\infty }e^{-x^{4}}\,dx={\frac {\sqrt {2\varpi {\sqrt {2\pi }}}}{4}},\quad {\text{analogous to}}\,\int _{0}^{\infty }e^{-x^{2}}\,dx={\frac {\sqrt {\pi }}{2}},$ a form of Gaussian integral.

The lemniscate constant appears in the evaluation of the integrals

${\frac {\pi }{\varpi }}=\int _{0}^{\frac {\pi }{2}}{\sqrt {\sin(x)}}\,dx=\int _{0}^{\frac {\pi }{2}}{\sqrt {\cos(x)}}\,dx$

${\frac {\varpi }{\pi }}=\int _{0}^{\infty }{\frac {dx}{\sqrt {\cosh(\pi x)}}}$

John Todd's lemniscate constants are defined by integrals:^[9]

$A=\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}$

$B=\int _{0}^{1}{\frac {x^{2}\,dx}{\sqrt {1-x^{4}}}}$

Circumference of an ellipse

The lemniscate constant satisfies the equation^[49]

${\frac {\pi }{\varpi }}=2\int _{0}^{1}{\frac {x^{2}\,dx}{\sqrt {1-x^{4}}}}$

Euler discovered in 1738 that for the rectangular elastica (first and second lemniscate constants)^[50]^[49]

${\textrm {arc}}\ {\textrm {length}}\cdot {\textrm {height}}=A\cdot B=\int _{0}^{1}{\frac {\mathrm {d} x}{\sqrt {1-x^{4}}}}\cdot \int _{0}^{1}{\frac {x^{2}\mathop {\mathrm {d} x} }{\sqrt {1-x^{4}}}}={\frac {\varpi }{2}}\cdot {\frac {\pi }{2\varpi }}={\frac {\pi }{4}}$

Now considering the circumference $C$ of the ellipse with axes ${\sqrt {2}}$ and $1$ , satisfying $2x^{2}+4y^{2}=1$ , Stirling noted that^[51]

${\frac {C}{2}}=\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}+\int _{0}^{1}{\frac {x^{2}\,dx}{\sqrt {1-x^{4}}}}$

Hence the full circumference is

$C={\frac {\pi }{\varpi }}+\varpi =3.820197789\ldots$

This is also the arc length of the sine curve on half a period:^[52]

$C=\int _{0}^{\pi }{\sqrt {1+\cos ^{2}(x)}}\,dx$

Other limits

Analogously to $2\pi =\lim _{n\to \infty }\left|{\frac {(2n)!}{\mathrm {B} _{2n}}}\right|^{\frac {1}{2n}}$ where $\mathrm {B} _{n}$ are Bernoulli numbers, we have $2\varpi =\lim _{n\to \infty }\left({\frac {(4n)!}{\mathrm {H} _{4n}}}\right)^{\frac {1}{4n}}$ where $\mathrm {H} _{n}$ are Hurwitz numbers.

Notes

^
See:
- Gauss, C. F. (1866). Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. p. 404
- Cox 1984, p. 281
- Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 199
- Bottazzini, Umberto; Gray, Jeremy (2013). Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory. Springer. doi:10.1007/978-1-4614-5725-1. ISBN 978-1-4614-5724-4. p. 57
- Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. ISBN 978-4-431-54918-5. p. 203
^
See:
- Finch 2003, p. 420
- Kobayashi, Hiroyuki; Takeuchi, Shingo (2019), "Applications of generalized trigonometric functions with two parameters", Communications on Pure & Applied Analysis, 18 (3): 1509–1521, arXiv:1903.07407, doi:10.3934/cpaa.2019072, S2CID 102487670
- Asai, Tetsuya (2007), Elliptic Gauss Sums and Hecke L-values at s=1, arXiv:0707.3711
- "A062539 - Oeis".
^ "A064853 - Oeis".
^ "Lemniscate Constant".
^ "Records set by y-cruncher". numberworld.org. Retrieved 2024-08-20.
^ "A014549 - Oeis".
^ Finch 2003, p. 420.
^ Neither of these proofs was rigorous from the modern point of view. See Cox 1984, p. 281
^ ^a ^b ^c Todd, John (January 1975). "The lemniscate constants". Communications of the ACM. 18 (1): 14–19. doi:10.1145/360569.360580. S2CID 85873.
^ ^a ^b "A085565 - Oeis". and "A076390 - Oeis".
^ Carlson, B. C. (2010), "Elliptic Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
^ In particular, Siegel proved that if $\operatorname {G} _{4}(\omega _{1},\omega _{2})$ and $\operatorname {G} _{6}(\omega _{1},\omega _{2})$ with $\operatorname {Im} (\omega _{2}/\omega _{1})>0$ are algebraic, then $\omega _{1}$ or $\omega _{2}$ is transcendental. Here, $\operatorname {G} _{4}$ and $\operatorname {G} _{6}$ are Eisenstein series. The fact that $\varpi$ is transcendental follows from $\operatorname {G} _{4}(\varpi ,\varpi i)=1/15$ and $\operatorname {G} _{6}(\varpi ,\varpi i)=0.$
Apostol, T. M. (1990). Modular Functions and Dirichlet Series in Number Theory (Second ed.). Springer. p. 12. ISBN 0-387-97127-0.
Siegel, C. L. (1932). "Über die Perioden elliptischer Funktionen". Journal für die reine und angewandte Mathematik (in German). 167: 62–69.
^ In particular, Schneider proved that the beta function $\mathrm {B} (a,b)$ is transcendental for all $a,b\in \mathbb {Q} \setminus \mathbb {Z}$ such that $a+b\notin \mathbb {Z} _{0}^{-}$ . The fact that $\varpi$ is transcendental follows from $\varpi ={\tfrac {1}{2}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )}$ and similarly for $B$ and $G$ from $\mathrm {B} {\bigl (}{\tfrac {1}{2}},{\tfrac {3}{4}}{\bigr )}.$
Schneider, Theodor (1941). "Zur Theorie der Abelschen Funktionen und Integrale". Journal für die reine und angewandte Mathematik. 183 (19): 110–128. doi:10.1515/crll.1941.183.110. S2CID 118624331.
^ G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis, Notices of the AMS 22, 1975, p. A-486
^ G. V. Chudnovsky: Contributions to The Theory of Transcendental Numbers, American Mathematical Society, 1984, p. 6
^ In fact, $\pi =2{\sqrt {2}}{\frac {M^{3}\left(1,{\frac {1}{\sqrt {2}}}\right)}{M'\left(1,{\frac {1}{\sqrt {2}}}\right)}}={\frac {1}{G^{3}M'\left(1,{\frac {1}{\sqrt {2}}}\right)}}.$
Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 45
^ Nesterenko, Y. V.; Philippon, P. (2001). Introduction to Algebraic Independence Theory. Springer. p. 27. ISBN 3-540-41496-7.
^
See:
- Cox 1984, p. 281
- Finch 2003, pp. 420–422
- Schappacher, Norbert (1997). "Some milestones of lemniscatomy" (PDF). In Sertöz, S. (ed.). Algebraic Geometry (Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey). Marcel Dekker. pp. 257–290.
^ Cox 1984, p. 277.
^ "A113847 - Oeis".
^ Cremona, J. E. (1997). Algorithms for Modular Elliptic Curves (2nd ed.). Cambridge University Press. ISBN 0521598206. p. 31, formula (2.8.10)
^ In fact, the series ${\textstyle \sum _{n=1}^{\infty }{\frac {\nu (n)}{n^{s}}}}$ converges for $\operatorname {\Re } s>5/6$ .
^ Murty, Vijaya Kumar (1995). Seminar on Fermat's Last Theorem. American Mathematical Society. p. 16. ISBN 9780821803134.
^ Cohen, Henri (1993). A Course in Computational Algebraic Number Theory. Springer-Verlag. p. 382–406. ISBN 978-3-642-08142-2.
^ "Elliptic curve with LMFDB label 32.a3 (Cremona label 32a2)". The L-functions and modular forms database.
^ The function $F$ is the unique weight $2$ level $32$ new form and it satisfies the functional equation
$F\left(-{\frac {1}{\tau }}\right)=-{\frac {\tau ^{2}}{32}}F\left({\frac {\tau {\vphantom {1}}}{32}}\right).$
^ The $\nu$ function is closely related to the $\xi$ function which is the multiplicative function defined by
$\xi (p^{n})={\begin{cases}{\mathcal {N}}_{p}',&p\in \mathbb {P} ,\,n=1\\[5mu]\xi (p^{n-1})+\chi (p)^{n},&p\in \mathbb {P} ,\,n\geq 2\end{cases}}$
where ${\mathcal {N}}_{p}'$ is the number of solutions of the equation
$a^{2}+b^{2}=p,\quad p\in \mathbb {P}$
in variables $a,b$ that are non-negative integers (see Fermat's theorem on sums of two squares) and $\chi$ is the Dirichlet character from the Leibniz formula for π; also
$\sum _{d|n}\chi (d)=\xi (n)$
for any positive integer $n$ where the sum extends only over positive divisors; the relation between $\nu$ and $\xi$ is
$\sum _{k=0}^{n}(-1)^{k}\xi (4k+1)\xi (4n-4k+1)=\nu (2n+1)$
where $n$ is any non-negative integer.
^ The $\nu$ function also appears in
$\sum _{z\in \mathbb {G} ;\,z{\overline {z}}=n}z=\nu (n)$
where $n$ is any positive integer and $\operatorname {\mathbb {G} }$ is the set of all Gaussian integers of the form
$(-1)^{\frac {a\pm b-1}{2}}(a\pm bi)$
where $a$ is odd and $b$ is even. The $\xi$ function from the previous note satisfies
$\left|\{z:z\in \mathbb {G} \land z{\overline {z}}=n\}\right|=\xi (n)$
where $n$ is positive odd.
^ Rubin, Karl (1987). "Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication". Inventiones mathematicae. 89: 528.
^ "Newform orbit 1.12.a.a". The L-functions and modular forms database.
^ Levin (2006)
^ Hyde (2014) proves the validity of a more general Wallis-like formula for clover curves; here the special case of the lemniscate is slightly transformed, for clarity.
^ Hyde, Trevor (2014). "A Wallis product on clovers" (PDF). The American Mathematical Monthly. 121 (3): 237–243. doi:10.4169/amer.math.monthly.121.03.237. S2CID 34819500.
^ Bottazzini, Umberto; Gray, Jeremy (2013). Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory. Springer. doi:10.1007/978-1-4614-5725-1. ISBN 978-1-4614-5724-4. p. 60
^ Todd (1975)
^ Cox 1984, p. 307, eq. 2.21 for the first equality. The second equality can be proved by using the pentagonal number theorem.
^ Berndt, Bruce C. (1998). Ramanujan's Notebooks Part V. Springer. ISBN 978-1-4612-7221-2. p. 326
^ This formula can be proved by hypergeometric inversion: Let
$\operatorname {a} (q)=\sum _{m,n\in \mathbb {Z} }q^{m^{2}+mn+n^{2}}$
where $q\in \mathbb {C}$ with $\left|q\right|<1$ . Then
$\operatorname {a} (q)={}_{2}F_{1}\left({\frac {1}{3}},{\frac {2}{3}},1,z\right)$
where
$q=\exp \left(-{\frac {2\pi }{\sqrt {3}}}{\frac {{}_{2}F_{1}(1/3,2/3,1,1-z)}{{}_{2}F_{1}(1/3,2/3,1,z)}}\right)$
where $z\in \mathbb {C} \setminus \{0,1\}$ . The formula in question follows from setting ${\textstyle z={\tfrac {1}{4}}{\bigl (}3{\sqrt {3}}-5{\bigr )}}$ .
^ Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 232
^ Garrett, Paul. "Level-one elliptic modular forms" (PDF). University of Minnesota. p. 11—13
^ The formula follows from the hypergeometric transformation
${}_{3}F_{2}\left({\frac {1}{4}},{\frac {1}{2}},{\frac {3}{4}},1,1,16z{\frac {(1-z)^{2}}{(1+z)^{4}}}\right)=(1+z)\,{}_{2}F_{1}\left({\frac {1}{2}},{\frac {1}{2}},1,z\right)^{2}$
where $z=\lambda (1+5i)$ and $\lambda$ is the modular lambda function.
^ Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press. ISBN 978-0-521-85419-1. p. 140 (eq. 3.34), p. 153. There's an error on p. 153: $4[\Gamma (3+s/4)/\Gamma (1+s/4)]^{2}$ should be $4[\Gamma ((3+s)/4)/\Gamma ((1+s)/4)]^{2}$ .
^ Khrushchev, Sergey (2008). Orthogonal Polynomials and Continued Fractions (First ed.). Cambridge University Press. ISBN 978-0-521-85419-1. p. 146, 155
^ Perron, Oskar (1957). Die Lehre von den Kettenbrüchen: Band II (in German) (Third ed.). B. G. Teubner. p. 36, eq. 24
^ "A062540 - OEIS". oeis.org. Retrieved 2022-09-14.
^ "A053002 - OEIS". oeis.org.
^ Blagouchine, Iaroslav V. (2014). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". The Ramanujan Journal. 35 (1): 21–110. doi:10.1007/s11139-013-9528-5. S2CID 120943474.
^ "A068467 - Oeis".
^ ^a ^b Cox 1984, p. 313.
^ Levien (2008)
^ Cox 1984, p. 312.
^ Adlaj, Semjon (2012). "An Eloquent Formula for the Perimeter of an Ellipse" (PDF). American Mathematical Society. p. 1097. One might also observe that the length of the "sine" curve over half a period, that is, the length of the graph of the function sin(t) from the point where t = 0 to the point where t = π , is ${\sqrt {2}}l(1/{\sqrt {2}})=L+M$ . In this paper $M=1/G=\pi /\varpi$ and $L=\pi /M=G\pi =\varpi$ .