Ratio of the perimeter of Bernoulli's lemniscate to its diameter
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. Equivalently, the perimeter of the lemniscate formula_1 is 2"ϖ". The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755. It also appears in evaluation of the gamma and beta function at certain rational values. The symbol ϖ is a cursive variant of π known as variant pi represented in Unicode by the character .
Sometimes the quantities 2"ϖ" or "ϖ/2" are referred to as "the" lemniscate constant. 
As of 2024 over 1.2 trillion digits of this constant have been calculated.
History.
Gauss's constant, denoted by "G", is equal to "ϖ" /π ≈ 0.8346268 and named after Carl Friedrich Gauss, who calculated it via the arithmetic–geometric mean as formula_2. By 1799, Gauss had two proofs of the theorem that formula_3 where formula_4 is the lemniscate constant.
John Todd named two more lemniscate constants, the "first lemniscate constant" "A" = "ϖ"/2 ≈ 1.3110287771 and the "second lemniscate constant" "B" = π/(2"ϖ") ≈ 0.5990701173.
The lemniscate constant formula_4 and Todd's first lemniscate constant formula_6 were proven transcendental by Carl Ludwig Siegel in 1932 and later by Theodor Schneider in 1937 and Todd's second lemniscate constant formula_7 and Gauss's constant formula_8 were proven transcendental by Theodor Schneider in 1941. In 1975, Gregory Chudnovsky proved that the set formula_9 is algebraically independent over formula_10, which implies that formula_6 and formula_7 are algebraically independent as well. But the set formula_13 (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over formula_10. In 1996, Yuri Nesterenko proved that the set formula_15 is algebraically independent over formula_10.
Forms.
Usually, formula_4 is defined by the first equality below, but it has many equivalent forms:
formula_18
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 formula_19,
formula_20
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 formula_21 published in 1800:formula_22John Todd's lemniscate constants may be given in terms of the beta function B:
formula_23
As a special value of L-functions.
formula_24
which is analogous to
formula_25
where formula_26 is the Dirichlet beta function and formula_27 is the Riemann zeta function.
Analogously to the Leibniz formula for π,
formula_28
we have
formula_29
where formula_30 is the L-function of the elliptic curve formula_31 over formula_10; this means that formula_33 is the multiplicative function given by
formula_34
where formula_35 is the number of solutions of the congruence
formula_36
in variables formula_37 that are non-negative integers (formula_38 is the set of all primes).
Equivalently, formula_33 is given by
formula_40
where formula_41 such that formula_42 and formula_43 is the eta function.
The above result can be equivalently written as
formula_44
(the number formula_45 is the conductor of formula_46) and also tells us that the BSD conjecture is true for the above formula_46.
The first few values of formula_33 are given by the following table; if formula_49 such that formula_50 doesn't appear in the table, then formula_51:
formula_52
As a special value of other functions.
Let formula_53 be the minimal weight level formula_54 new form. Then
formula_55
The formula_56-coefficient of formula_53 is the Ramanujan tau function.
Series.
Viète's formula for π can be written:
formula_58
An analogous formula for ϖ is:
formula_59
The Wallis product for π is:
formula_60
An analogous formula for ϖ is:
formula_61
A related result for Gauss's constant (formula_62) is:
formula_63
An infinite series discovered by Gauss is:
formula_64
The Machin formula for π is formula_65 and several similar formulas for π can be developed using trigonometric angle sum identities, e.g. Euler's formula formula_66. Analogous formulas can be developed for ϖ, including the following found by Gauss: formula_67, where formula_68 is the lemniscate arcsine.
The lemniscate constant can be rapidly computed by the series
formula_69
where formula_70 (these are the generalized pentagonal numbers). Also
formula_71
Other limits.
Analogously to
formula_72
where formula_73 are Bernoulli numbers, we have
formula_74
where formula_75 are Hurwitz numbers.
Notes.
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