lemniscate constant

Gauss's constant, denoted by G, is equal to {{math|ϖ /{{pi}} ≈ 0.8346268}}{{cite web |title=A014549 - Oeis |url=http://oeis.org/A014549}} and named after Carl Friedrich Gauss, who calculated it via the arithmetic–geometric mean as $1/M\backslash bigl(1,\backslash sqrt\{2\}\backslash bigr)$.{{sfn|Finch|2003|p=420}} By 1799, Gauss had two proofs of the theorem that $M\backslash bigl(1,\backslash sqrt2\backslash bigr)=\backslash pi/\backslash varpi$ where $\backslash varpi$ is the lemniscate constant.Neither of these proofs was rigorous from the modern point of view. See {{harvnb|Cox|1984|p=281}}

John Todd named two more lemniscate constants, the first lemniscate constant {{math|1=A = ϖ/2 ≈ 1.3110287771}} and the second lemniscate constant {{math|1=B = {{pi}}/(2ϖ) ≈ 0.5990701173}}.{{Cite journal |last=Todd |first=John |date=January 1975 |title=The lemniscate constants |journal=Communications of the ACM |volume=18 |pages=14–19 |doi=10.1145/360569.360580 |s2cid=85873 |doi-access=free |number=1}}{{Cite web |title=A085565 - Oeis |url=http://oeis.org/A085565}} and {{cite web |title=A076390 - Oeis |url=http://oeis.org/A076390}}{{dlmf|first=B. C.|last=Carlson|id=19.20.E2|title=Elliptic Integrals}}

The lemniscate constant $\backslash varpi$ 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.{{r|todd}} In particular, Siegel proved that if $\backslash operatorname\{G\}\_4(\backslash omega\_1,\backslash omega\_2)$ and $\backslash operatorname\{G\}\_6(\backslash omega\_1,\backslash omega\_2)$ with $\backslash operatorname\{Im\}(\backslash omega\_2/\backslash omega\_1)>0$ are algebraic, then $\backslash omega\_1$ or $\backslash omega\_2$ is transcendental. Here, $\backslash operatorname\{G\}\_4$ and $\backslash operatorname\{G\}\_6$ are Eisenstein series. The fact that $\backslash varpi$ is transcendental follows from $\backslash operatorname\{G\}\_4(\backslash varpi,\backslash varpi\; i)=1/15$ and $\backslash operatorname\{G\}\_6(\backslash varpi,\backslash varpi\; i)=0.$ {{pb}} {{Cite book |last=Apostol |first=T. M. |title=Modular Functions and Dirichlet Series in Number Theory|publisher=Springer |year=1990 |isbn=0-387-97127-0|page=12|edition=Second}} {{pb}} {{Cite journal |last=Siegel |first=C. L. |date=1932 |title=Über die Perioden elliptischer Funktionen. |url=https://eudml.org/doc/149791 |journal=Journal für die reine und angewandte Mathematik |volume=167 |pages=62–69|doi=10.1515/crll.1932.167.62 |language=German}}In particular, Schneider proved that the beta function $\backslash Beta\; (a,b)$ is transcendental for all $a,b\backslash in\backslash mathbb\{Q\}\backslash setminus\backslash mathbb\{Z\}$ such that $a+b\backslash notin\; \backslash mathbb\{Z\}\_0^-$. The fact that $\backslash varpi$ is transcendental follows from $\backslash varpi=\backslash tfrac\{1\}\{2\}\backslash Beta\; \backslash bigl(\backslash tfrac\{1\}\{4\},\backslash tfrac\{1\}\{2\}\backslash bigr)$ and similarly for {{mvar|B}} and {{mvar|G}} from $\backslash Beta\; \backslash bigl(\backslash tfrac\{1\}\{2\},\backslash tfrac\{3\}\{4\}\backslash bigr).$ {{pb}} {{cite journal |first=Theodor |last=Schneider |url=https://www.deepdyve.com/lp/de-gruyter/zur-theorie-der-abelschen-funktionen-und-integrale-mn0U50bvkB |title=Zur Theorie der Abelschen Funktionen und Integrale |year=1941 |journal=Journal für die reine und angewandte Mathematik |volume=183 |number=19 |pages=110–128 |doi=10.1515/crll.1941.183.110 |s2cid=118624331 }} In 1975, Gregory Chudnovsky proved that the set $\backslash \{\backslash pi,\backslash varpi\backslash \}$ is algebraically independent over $\backslash mathbb\{Q\}$, which implies that $A$ and $B$ are algebraically independent as well.G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis, Notices of the AMS 22, 1975, p. A-486G. V. Chudnovsky: Contributions to The Theory of Transcendental Numbers, American Mathematical Society, 1984, p. 6 But the set $\backslash bigl\backslash \{\backslash pi,M\backslash bigl(1,1/\backslash sqrt\{2\}\backslash bigr),M\text{'}\backslash bigl(1,1/\backslash sqrt\{2\}\backslash bigr)\backslash bigr\backslash \}$ (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over $\backslash mathbb\{Q\}$.In fact, $$\backslash pi=2\backslash sqrt\{2\}\backslash frac\{M^3\backslash left(1,\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash right)\}\{M\text{'}\backslash left(1,\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash right)\}=\backslash frac\{1\}\{G^3\; M\text{'}\backslash left(1,\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash right)\}.$$ {{pb}} {{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 45 In 1996, Yuri Nesterenko proved that the set $\backslash \{\backslash pi,\backslash varpi,e^\{\backslash pi\}\backslash \}$ is algebraically independent over $\backslash mathbb\{Q\}$.{{Cite book |last1=Nesterenko |first1=Y. V. |last2=Philippon|first2=P.|title=Introduction to Algebraic Independence Theory|publisher=Springer |year=2001 |isbn=3-540-41496-7|page=27}}

Usually, $\backslash varpi$ is defined by the first equality below, but it has many equivalent forms:See:

• {{harvnb|Cox|1984|p=281}}
• {{harvnb|Finch|2003|pp=420–422}}
• {{cite book  |chapter=Some milestones of lemniscatomy  |last1=Schappacher |first1=Norbert |author-link1=Norbert Schappacher | date= 1997 |editor1-last=Sertöz |editor1-first=S. |title=Algebraic Geometry |type=Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey |publisher=Marcel Dekker |pages=257–290 | chapter-url=http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1997_LemniscProvis.pdf}}

$$\backslash begin\{aligned\}$$

\varpi

&= 2\int_0^1\frac{\mathrm{d}t}{\sqrt{1-t^4}}

= \sqrt2\int_0^\infty\frac{\mathrm{d}t}{\sqrt{1+t^4}}

= \int_0^1\frac{\mathrm dt}{\sqrt{t-t^3}}

= \int_1^\infty \frac{\mathrm dt}{\sqrt{t^3-t}}\\[6mu]

&= 4\int_0^\infty\Bigl(\sqrt[4]{1+t^{4}}-t\Bigr)\,\mathrm{d}t

= 2\sqrt2\int_0^1 \sqrt[4]{1-t^{4}}\mathop{\mathrm{d}t} =3\int_0^1 \sqrt{1-t^4}\,\mathrm dt\\[2mu]

&= 2K(i)

= \tfrac{1}{2}\Beta\bigl( \tfrac14, \tfrac12\bigr)

= \tfrac{1}{2\sqrt2}\Beta\bigl( \tfrac14, \tfrac14\bigr)

= \frac{\Gamma (1/4)^2}{2\sqrt{2\pi}}

= \frac{2-\sqrt2}{4}\frac{\zeta(3/4)^2}{\zeta(1/4)^2}\\[5mu]

&= 2.62205\;75542\;92119\;81046\;48395\;89891\;11941\ldots,

\end{aligned}

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

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

$$\backslash varpi=\backslash frac\{\backslash pi\}\{M\backslash bigl(1,\backslash sqrt2\backslash bigr)\}.$$

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 $M\backslash bigl(1,\; \backslash sqrt2\backslash bigr)$ published in 1800:{{sfn|Cox|1984|p=277}}$$G\; =\; \backslash frac\{1\}\{M\backslash bigl(1,\; \backslash sqrt2\backslash bigr)\}$$John Todd's lemniscate constants may be given in terms of the beta function B:

$$\backslash begin\{aligned\}$$

A &= \frac\varpi2 = \tfrac14 \Beta \bigl(\tfrac14,\tfrac12\bigr), \\[3mu]

B &= \frac{\pi}{2\varpi} =\tfrac14\Beta \bigl(\tfrac12,\tfrac34\bigr).

\end{aligned}

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

which is analogous to

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

where $\backslash beta$ is the Dirichlet beta function and $\backslash zeta$ is the Riemann zeta function.{{cite web | url=http://oeis.org/A113847 | title=A113847 - Oeis }}

Analogously to the Leibniz formula for π,

$$\backslash beta\; (1)=\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash chi\; (n)\}\{n\}=\backslash frac\{\backslash pi\}\{4\},$$

we have{{cite book |last=Cremona |first=J. E. |date= 1997 |edition=2nd|title=Algorithms for Modular Elliptic Curves|url=https://books.google.com/books?id=MtM8AAAAIAAJ|publisher=Cambridge University Press|isbn=0521598206}} p. 31, formula (2.8.10)In fact, the series $\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash nu\; (n)\}\{n^s\}$ converges for $\backslash operatorname\{\backslash Re\}s>5/6$.{{cite book |last=Murty |first=Vijaya Kumar |author-link=V. Kumar Murty|date= 1995 |page=16 |title=Seminar on Fermat's Last Theorem |publisher= American Mathematical Society|isbn=9780821803134}}{{Cite book| last=Cohen| first=Henri|author-link=Henri Cohen (number theorist)|date=1993|pages=382–406|title=A Course in Computational Algebraic Number Theory|publisher=Springer-Verlag|isbn=978-3-642-08142-2}}{{cite web |url=https://www.lmfdb.org/EllipticCurve/Q/32/a/3 |title=Elliptic curve with LMFDB label 32.a3 (Cremona label 32a2)|website=The L-functions and modular forms database}}

$$L(E,1)=\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash nu\; (n)\}\{n\}=\backslash frac\{\backslash varpi\}\{4\}$$

where $L$ is the L-function of the elliptic curve $E:\backslash ,\; y^2=x^3-x$ over $\backslash mathbb\{Q\}$; this means that $\backslash nu$ is the multiplicative function given by

$$\backslash nu\; (p^n)=\backslash begin\{cases\}$$

p - \mathcal{N}_p, & p\in\mathbb{P}, \, n=1 \\[5mu]

0, & p=2,\, n\ge 2 \\[5mu]

\nu (p)\nu (p^{n-1})-p\nu (p^{n-2}), & p\in\mathbb{P}\setminus\{2\},\, n\ge 2

\end{cases}

where $\backslash mathcal\{N\}\_p$ is the number of solutions of the congruence

$$a^3-a\backslash equiv\; b^2\; \backslash ,(\backslash operatorname\{mod\}p),\backslash quad\; p\backslash in\backslash mathbb\{P\}$$

in variables $a,b$ that are non-negative integers ($\backslash mathbb\{P\}$ is the set of all primes).

Equivalently, $\backslash nu$ is given by

$$F(\backslash tau)=\backslash eta\; (4\backslash tau)^2\backslash eta\; (8\backslash tau)^2=\backslash sum\_\{n=1\}^\backslash infty\; \backslash nu\; (n)\; q^n,\backslash quad\; q=e^\{2\backslash pi\; i\backslash tau\}$$

where $\backslash tau\backslash in\backslash mathbb\{C\}$ such that $\backslash operatorname\{\backslash Im\}\backslash tau\; >0$ and $\backslash eta$ is the eta function.The function $F$ is the unique weight $2$ level $32$ new form and it satisfies the functional equation

:$F\backslash left(-\backslash frac\{1\}\{\backslash tau\}\backslash right)=-\backslash frac\{\backslash tau^2\}\{32\}F\backslash left(\backslash frac\{\backslash tau\backslash vphantom1\}\{32\}\backslash right).$The $\backslash nu$ function is closely related to the $\backslash xi$ function which is the multiplicative function defined by

:$\backslash xi\; (p^n)\; =\; \backslash begin\{cases\}$

\mathcal{N}_p', & p\in\mathbb{P},\, n=1 \\[5mu]

\xi (p^{n-1}) + \chi (p)^n, & p\in\mathbb{P},\, n\ge 2

\end{cases}

where $\backslash mathcal\{N\}\_p\text{'}$ is the number of solutions of the equation

:$a^2+b^2=p,\backslash quad\; p\backslash in\backslash mathbb\{P\}$

in variables $a,b$ that are non-negative integers (see Fermat's theorem on sums of two squares) and $\backslash chi$ is the Dirichlet character from the Leibniz formula for π; also

:$\backslash sum\_\{d|n\}\backslash chi\; (d)=\backslash xi\; (n)$

for any positive integer $n$ where the sum extends only over positive divisors; the relation between $\backslash nu$ and $\backslash xi$ is

:$\backslash sum\_\{k=0\}^n\; (-1)^k\; \backslash xi\; (4k+1)\backslash xi\; (4n-4k+1)=\backslash nu\; (2n+1)$

where $n$ is any non-negative integer.The $\backslash nu$ function also appears in

:$\backslash sum\_\{z\backslash in\backslash mathbb\{G\};\backslash ,\; z\backslash overline\{z\}=n\}z=\backslash nu\; (n)$

where $n$ is any positive integer and $\backslash operatorname\{\backslash mathbb\{G\}\}$ is the set of all Gaussian integers of the form

:$(-1)^\{\backslash frac\{a\backslash pm\; b-1\}\{2\}\}(a\backslash pm\; bi)$

where $a$ is odd and $b$ is even. The $\backslash xi$ function from the previous note satisfies

:$\backslash left|\backslash \{z:z\backslash in\backslash mathbb\{G\}\backslash and\; z\backslash overline\{z\}=n\backslash \}\backslash right|=\backslash xi\; (n)$

where $n$ is positive odd.

The above result can be equivalently written as

$$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash nu\; (n)\}\{n\}e^\{-2\backslash pi\; n/\backslash sqrt\{32\}\}=\backslash frac\{\backslash varpi\}\{8\}$$

(the number $32$ is the conductor of $E$) and also tells us that the BSD conjecture is true for the above $E$.{{Cite journal |last=Rubin |first=Karl |date=1987 |title=Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication |url=https://eudml.org/doc/143493 |journal=Inventiones Mathematicae |volume=89 |issue=3 |page=528|doi=10.1007/BF01388984 |bibcode=1987InMat..89..527R }}

The first few values of $\backslash nu$ are given by the following table; if $1\backslash le\; n\backslash le\; 113$ such that $n$ doesn't appear in the table, then $\backslash nu\; (n)=0$:

$$\backslash begin\{array\}\{r|r|r|r\}$$

n & \nu (n) & n & \nu (n) \\

\hline

1 & 1 & 53 & 14 \\

5 & -2 & 61 & -10 \\

9 & -3 & 65 & -12 \\

13 & 6 & 73 & -6 \\

17 & 2 & 81 & 9 \\

25 & -1 & 85 & -4 \\

29 & -10 & 89 & 10\\

37 & -2 & 97 & 18 \\

41 & 10 & 101 & -2 \\

45 & 6 & 109 & 6 \\

49 & -7 & 113 & -14 \\

\end{array}

Let $\backslash Delta$ be the minimal weight level $1$ new form. Then{{cite web|url=https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/1/12/a/a/|title=Newform orbit 1.12.a.a|website=The L-functions and modular forms database}}

$$\backslash Delta\; (i)=\backslash frac\{1\}\{64\}\backslash left(\backslash frac\{\backslash varpi\}\{\backslash pi\}\backslash right)^\{12\}.$$

The $q$-coefficient of $\backslash Delta$ is the Ramanujan tau function.

Viète's formula for {{mvar|π}} can be written:

\frac2\pi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12\sqrt{\frac12 + \frac12\sqrt\frac12}} \cdots

An analogous formula for {{mvar|ϖ}} is:Levin (2006)

\frac2\varpi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12 \bigg/ \!\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12 \Bigg/ \!\sqrt{\frac12 + \frac12 \bigg/ \!\sqrt\frac12}} \cdots

The Wallis product for {{mvar|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 {{mvar|ϖ}} is: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.

\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 ($G=\backslash varpi\; /\; \backslash pi$) is:{{cite journal |last1=Hyde |first1=Trevor |year=2014 |title=A Wallis product on clovers |journal=The American Mathematical Monthly |volume=121 |issue=3 |pages=237–243 |url=https://math.uchicago.edu/~tghyde/Hyde%20--%20A%20Wallis%20product%20on%20clovers.pdf |doi=10.4169/amer.math.monthly.121.03.237 |s2cid=34819500 }}

\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:{{cite book |last1=Bottazzini |first1=Umberto |authorlink1=Umberto Bottazzini |last2=Gray |first2=Jeremy |authorlink2=Jeremy Gray |date=2013 |title=Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory |publisher=Springer |doi=10.1007/978-1-4614-5725-1 |isbn=978-1-4614-5724-4 }} p. 60

\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\cdot3^2}{2^2\cdot4^2} - \frac{1^2\cdot3^2\cdot5^2}{2^2\cdot4^2\cdot6^2} + \cdots

The Machin formula for {{mvar|π}} is $\backslash tfrac14\backslash pi\; =\; 4\; \backslash arctan\; \backslash tfrac15\; -\; \backslash arctan\; \backslash tfrac1\{239\},$ and several similar formulas for {{mvar|π}} can be developed using trigonometric angle sum identities, e.g. Euler's formula $\backslash tfrac14\backslash pi\; =\; \backslash arctan\backslash tfrac12\; +\; \backslash arctan\backslash tfrac13$. Analogous formulas can be developed for {{mvar|ϖ}}, including the following found by Gauss: $\backslash tfrac12\backslash varpi\; =\; 2\; \backslash operatorname\{arcsl\}\; \backslash tfrac12\; +\; \backslash operatorname\{arcsl\}\; \backslash tfrac7\{23\}$, where $\backslash operatorname\{arcsl\}$ is the lemniscate arcsine.Todd (1975)

The lemniscate constant can be rapidly computed by the series{{harvnb|Cox|1984|p=307|loc=eq. 2.21}} for the first equality. The second equality can be proved by using the pentagonal number theorem.{{Cite book |last1=Berndt |first1=Bruce C. |title=Ramanujan's Notebooks Part V |publisher=Springer |year=1998 |isbn=978-1-4612-7221-2}} p. 326

:$\backslash varpi=2^\{-1/2\}\backslash pi\backslash biggl(\backslash sum\_\{n\backslash in\backslash mathbb\{Z\}\}e^\{-\backslash pi\; n^2\}\backslash biggr)^2=2^\{1/4\}\backslash pi\; e^\{-\backslash pi/12\}\; \backslash biggl(\backslash sum\_\{n\backslash in\backslash mathbb\{Z\}\}(-1)^n\; e^\{-\backslash pi\; p\_n\}\backslash biggr)^2$

where $p\_n=\backslash tfrac12(3n^2-n)$ (these are the generalized pentagonal numbers). AlsoThis formula can be proved by hypergeometric inversion: Let

:$\backslash operatorname\{a\}(q)=\backslash sum\_\{m,n\backslash in\backslash mathbb\{Z\}\}q^\{m^2+mn+n^2\}$

where $q\backslash in\backslash mathbb\{C\}$ with . Then

:$\backslash operatorname\{a\}(q)=\{\}\_2F\_1\backslash left(\backslash frac\{1\}\{3\},\backslash frac\{2\}\{3\},1,z\backslash right)$

where

:$q=\backslash exp\backslash left(-\backslash frac\{2\backslash pi\}\{\backslash sqrt\{3\}\}\backslash frac\{\{\}\_2F\_1(1/3,2/3,1,1-z)\}\{\{\}\_2F\_1(1/3,2/3,1,z)\}\backslash right)$

where $z\backslash in\backslash mathbb\{C\}\backslash setminus\backslash \{0,1\backslash \}$. The formula in question follows from setting $z=\backslash tfrac14\backslash bigl(3\backslash sqrt\{3\}-5\backslash bigr)$.

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

In a spirit similar to that of the Basel problem,

:$\backslash sum\_\{z\backslash in\backslash mathbb\{Z\}[i]\backslash setminus\backslash \{0\backslash \}\}\backslash frac\{1\}\{z^4\}=G\_4(i)=\backslash frac\{\backslash varpi\; ^4\}\{15\}$

where $\backslash mathbb\{Z\}[i]$ are the Gaussian integers and $G\_4$ is the Eisenstein series of weight {{tmath|4}} (see Lemniscate elliptic functions § Hurwitz numbers for a more general result).{{Cite book |last1=Eymard |first1=Pierre |last2=Lafon| first2=Jean-Pierre |title=The Number Pi |publisher=American Mathematical Society |year=2004 |isbn=0-8218-3246-8}} p. 232

A related result is

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash sigma\_3(n)e^\{-2\backslash pi\; n\}=\backslash frac\{\backslash varpi^4\}\{80\; \backslash pi^4\}-\backslash frac\{1\}\{240\}$

where $\backslash sigma\_3$ is the sum of positive divisors function.{{cite web |url=http://www-users.math.umn.edu/~garrett/m/mfms/notes_2015-16/10_level_one.pdf |title=Level-one elliptic modular forms |last=Garrett |first=Paul |website=University of Minnesota}} p. 11—13

In 1842, Malmsten found

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

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

The lemniscate constant is given by the rapidly converging series

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

The constant is also given by the infinite product

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

AlsoThe formula follows from the hypergeometric transformation

:$\{\}\_3F\_2\backslash left(\backslash frac14,\backslash frac12,\backslash frac34,1,1,16z\backslash frac\{(1-z)^2\}\{(1+z)^4\}\backslash right)=(1+z)\backslash ,\{\}\_2F\_1\backslash left(\backslash frac12,\backslash frac12,1,z\backslash right)^2$

where $z=\backslash lambda\; (1+5i)$ and $\backslash lambda$ is the modular lambda function.

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

A (generalized) continued fraction for {{mvar|π}} is

$$\backslash frac\backslash pi2=1\; +\; \backslash cfrac\{1\}\{1\; +\; \backslash cfrac\{1\backslash cdot\; 2\}\{1\; +\; \backslash cfrac\{2\backslash cdot\; 3\}\{1\; +\; \backslash cfrac\{3\backslash cdot\; 4\}\{1+\backslash ddots\}\}\}\}$$

An analogous formula for {{mvar|ϖ}} is{{r|lemniscate constant A}}

$$\backslash frac\backslash varpi2=\; 1\; +\; \backslash cfrac\{1\}\{2\; +\; \backslash cfrac\{2\backslash cdot\; 3\}\{2\; +\; \backslash cfrac\{4\backslash cdot\; 5\}\{2\; +\; \backslash cfrac\{6\backslash cdot\; 7\}\{2+\backslash ddots\}\}\}\}$$

Define Brouncker's continued fraction by{{Cite book |last1=Khrushchev |first1=Sergey |title=Orthogonal Polynomials and Continued Fractions |publisher=Cambridge University Press |year=2008 |edition=First |isbn=978-0-521-85419-1}} p. 140 (eq. 3.34), p. 153. There's an error on p. 153: $4\; [\backslash Gamma(3+s/4)/\backslash Gamma(1+s/4)]^2$ should be $4[\backslash Gamma((3+s)/4)/\backslash Gamma((1+s)/4)]^2$.

$$b(s)=s\; +\; \backslash cfrac\{1^2\}\{2s\; +\; \backslash cfrac\{3^2\}\{2s\; +\; \backslash cfrac\{5^2\}\{2s+\backslash ddots\}\}\},\backslash quad\; s>0.$$

Let $n\backslash ge\; 0$ except for the first equality where $n\backslash ge\; 1$. Then{{Cite book |last1=Khrushchev |first1=Sergey |title=Orthogonal Polynomials and Continued Fractions |publisher=Cambridge University Press |year=2008 |edition=First |isbn=978-0-521-85419-1}} p. 146, 155{{Cite book |last1=Perron |first1=Oskar |language=German |author-link=Oskar Perron|title=Die Lehre von den Kettenbrüchen: Band II |publisher=B. G. Teubner |year=1957 |edition=Third}} p. 36, eq. 24

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{align}

For example,

$$\backslash begin\{align\}$$

b(1) &= \frac{4}{\pi}, &

b(2) &= \frac{\varpi^2}{\pi}, &

b(3) &= \pi, &

b(4) &= \frac{9\pi}{\varpi^2}.

\end{align}

In fact, the values of $b(1)$ and $b(2)$, coupled with the functional equation

$$b(s+2)=\backslash frac\{(s+1)^2\}\{b(s)\},$$

determine the values of $b(n)$ for all $n$.

Simple continued fractions for the lemniscate constant and related constants include{{Cite web |title=A062540 - OEIS |url=http://oeis.org/A062540 |access-date=2022-09-14 |website=oeis.org}}{{Cite web |title=A053002 - OEIS|url=http://oeis.org/A053002|website=oeis.org}}

$$\backslash begin\{align\}$$

\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{align}

File:Lemniscate constant as an integral.png

The lemniscate constant {{mvar|ϖ}} is related to the area under the curve $x^4\; +\; y^4\; =\; 1$. Defining $\backslash pi\_n\; \backslash mathrel\{:=\}\; \backslash Beta\backslash bigl(\backslash tfrac1n,\; \backslash tfrac1n\; \backslash bigr)$, twice the area in the positive quadrant under the curve $x^n\; +\; y^n\; =\; 1$ is $2\; \backslash int\_0^1\; \backslash sqrt[n]\{1\; -\; x^n\}\backslash mathop\{\backslash mathrm\{d\}x\}\; =\; \backslash tfrac1n\; \backslash pi\_n.$ In the quartic case, $\backslash tfrac14\; \backslash pi\_4\; =\; \backslash tfrac1\backslash sqrt\{2\}\; \backslash varpi.$

In 1842, Malmsten discovered that{{cite journal |last1=Blagouchine |first1=Iaroslav V. |date=2014 |title=Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results |url=https://www.researchgate.net/publication/257381156 |journal=The Ramanujan Journal |volume=35 |issue=1 |pages=21–110|doi=10.1007/s11139-013-9528-5 |s2cid=120943474 }}

$$\backslash int\_0^1\; \backslash frac\{\backslash log\; (-\backslash log\; x)\}\{1+x^2\}\backslash ,\; dx=\backslash frac\{\backslash pi\}\{2\}\backslash log\backslash frac\{\backslash pi\}\{\backslash varpi\backslash sqrt\{2\}\}.$$

Furthermore,

$$\backslash int\_0^\backslash infty\; \backslash frac\{\backslash tanh\; x\}\{x\}e^\{-x\}\backslash ,\; dx=\backslash log\backslash frac\{\backslash varpi^2\}\{\backslash pi\}$$

and{{cite web | url=https://oeis.org/A068467 | title=A068467 - Oeis }}

$$\backslash int\_0^\backslash infty\; e^\{-x^4\}\backslash ,\; dx=\backslash frac\{\backslash sqrt\{2\backslash varpi\backslash sqrt\{2\backslash pi\}\}\}\{4\},\backslash quad\backslash text\{analogous\; to\}\backslash ,\backslash int\_0^\backslash infty\; e^\{-x^2\}\backslash ,\; dx=\backslash frac\{\backslash sqrt\{\backslash pi\}\}\{2\},$$

a form of Gaussian integral.

The lemniscate constant appears in the evaluation of the integrals

$$\{\backslash frac\{\backslash pi\}\{\backslash varpi\}\}\; =\; \backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\backslash sqrt\{\backslash sin(x)\}\backslash ,dx=\backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\backslash sqrt\{\backslash cos(x)\}\backslash ,dx$$

$$\backslash frac\{\backslash varpi\}\{\backslash pi\}\; =\; \backslash int\_0^\{\backslash infty\}\{\backslash frac\{dx\}\{\backslash sqrt\{\backslash cosh(\backslash pi\; x)\}\}\}$$

John Todd's lemniscate constants are defined by integrals:{{r|todd}}

$$A\; =\; \backslash int\_0^1\backslash frac\{dx\}\{\backslash sqrt\{1\; -\; x^4\}\}$$

$$B\; =\; \backslash int\_0^1\backslash frac\{x^2\backslash ,\; dx\}\{\backslash sqrt\{1\; -\; x^4\}\}$$

The lemniscate constant satisfies the equation{{sfn|Cox|1984|p=313}}

$$\backslash frac\{\backslash pi\}\{\backslash varpi\}\; =\; 2\; \backslash int\_0^1\backslash frac\{x^2\backslash ,\; dx\}\{\backslash sqrt\{1\; -\; x^4\}\}$$

Euler discovered in 1738 that for the rectangular elastica (first and second lemniscate constants)Levien (2008){{sfn|Cox|1984|p=313}}

$$\backslash textrm\{arc\}\backslash \; \backslash textrm\{length\}\backslash cdot\backslash textrm\{height\}\; =\; A\; \backslash cdot\; B\; =\; \backslash int\_0^1\; \backslash frac\{\backslash mathrm\{d\}x\}\{\backslash sqrt\{1\; -\; x^4\}\}\; \backslash cdot\; \backslash int\_0^1\; \backslash frac\{x^2\; \backslash mathop\{\backslash mathrm\{d\}x\}\}\{\backslash sqrt\{1\; -\; x^4\}\}\; =\; \backslash frac\backslash varpi2\; \backslash cdot\; \backslash frac\backslash pi\{2\backslash varpi\}\; =\; \backslash frac\backslash pi4$$

Now considering the circumference $C$ of the ellipse with axes $\backslash sqrt\{2\}$ and $1$, satisfying $2x^2\; +\; 4y^2\; =\; 1$, Stirling noted that{{sfn|Cox|1984|p=312}}

$$\backslash frac\{C\}\{2\}\; =\; \backslash int\_0^1\backslash frac\{dx\}\{\backslash sqrt\{1\; -\; x^4\}\}\; +\; \backslash int\_0^1\backslash frac\{x^2\backslash ,dx\}\{\backslash sqrt\{1\; -\; x^4\}\}$$

Hence the full circumference is

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

This is also the arc length of the sine curve on half a period:{{Cite web|url=https://www.ams.org/notices/201208/rtx120801094p.pdf|title=An Eloquent Formula for the Perimeter of an Ellipse|last=Adlaj|first=Semjon|date=2012|website=American Mathematical Society|page=1097|quote=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 {{tmath|1=\sin(t)}} from the point where {{tmath|1=t = 0}} to the point where {{tmath|1=t = \pi}} , is $\backslash sqrt\{2\}\; l(1/\backslash sqrt\{2\})\; =\; L\; +\; M$.}} In this paper $M=1/G=\backslash pi/\backslash varpi$ and $L\; =\; \backslash pi/M=G\backslash pi=\backslash varpi$.

$$C\; =\; \backslash int\_0^\backslash pi\; \backslash sqrt\{1+\backslash cos^2(x)\}\backslash ,dx$$

Analogously to

$$2\backslash pi=\backslash lim\_\{n\backslash to\backslash infty\}\backslash left|\backslash frac\{(2n)!\}\{\backslash mathrm\{B\}\_\{2n\}\}\backslash right|^\{\backslash frac\{1\}\{2n\}\}$$

where $\backslash mathrm\{B\}\_n$ are Bernoulli numbers, we have

$$2\backslash varpi=\backslash lim\_\{n\backslash to\backslash infty\}\backslash left(\backslash frac\{(4n)!\}\{\backslash mathrm\{H\}\_\{4n\}\}\backslash right)^\{\backslash frac\{1\}\{4n\}\}$$

where $\backslash mathrm\{H\}\_n$ are Hurwitz numbers.

{{Reflist}}

• {{mathworld|urlname=LemniscateConstant|title=Lemniscate Constant}}
• Sequences A014549, A053002, and [https://oeis.org/A062539 A062539] in OEIS
• {{cite journal |last1=Cox |first1=David A. |title=The Arithmetic-Geometric Mean of Gauss |journal=L'Enseignement Mathématique |date=January 1984 |volume=30|issue=2 |pages=275–330 |doi=10.5169/seals-53831 |url=https://webspace.science.uu.nl/~wepst101/elliptic/cox_agm.pdf |access-date=25 June 2022}}
• {{cite book |last1=Finch |first1=Steven R. |title=Mathematical Constants |date=18 August 2003 |publisher=Cambridge University Press |isbn=978-0-521-81805-6 |pages=420–422 |url=https://books.google.com/books?id=Pl5I2ZSI6uAC |language=en}}

• {{Cite web|date=2021-10-17|title=Gauss's constant and where it occurs|url=https://www.johndcook.com/blog/2021/10/17/gauss-constant/|website=www.johndcook.com|language=en-US}}

{{Irrational number}}

Category:Mathematical constants

Category:Real transcendental numbers