Let be Chebyshev constants. Schönhage (1973) proved that

(1)

It was conjectured that the number

(2)

The number is therefore known as the "one-ninth constant" or Halphen constant (Finch 2003, p. 261) and its reciprocal is sometimes known as Varga's constant. In 1981, N. Trefethen (Trefethen and Gutknecht 1983) refuted the conjecture that by computing Varga's constant as

(3)

(Sloane's A073007). Upon making this discovery on May 23, 1981, Trefethen (then a graduate student at Stanford University) was so excited that he sent a telegram to his coauthor M. Gutknecht in Zurich saying simply "9.28903?" Carpenter et al. (1984) subsequently confirmed this result by computing

(4)

(Sloane's A072558) numerically.

Gonchar and Rakhmanov showed that the limit exists and formally disproved the 1/9 conjecture in 1986, a result which Gonchar presented at the International Congress of Mathematicians in Berkeley, California. Magnus (1986, 1988) subsequently showed that is exactly given by

(5)

where is the complete elliptic integral of the first kind, and

(6)

(Sloane's A086199) is the parameter which solves

(7)

and is the complete elliptic integral of the second kind.

is also given by the unique positive root of

(8)

where

(9)

and

(10)

(Gonchar and Rakhmanov 1987). may also be computed by writing as

(11)

where and , then

(12)

(Gonchar and Rakhmanov 1987).

A generating function for is given by

			(13)
			(14)
			(15)

where and are complete elliptic integrals of the first and second kind, respectively, and the elliptic modulus is expressed in terms of the nome (M. Somos, pers. comm., Jul. 27, 2006).

Yet another equation for is due to Magnus (1988). is the unique solution with of

(16)

an equation which had been studied and whose root had been computed by Halphen in 1886. It has therefore been suggested (Varga 1990) that the constant be called the Halphen constant.

Carpenter, A. J.; Ruttan, A.; and Varga, R. S. "Extended Numerical Computations on the '' Conjecture in Rational Approximation Theory." In Rational Approximation and Interpolation (Tampa, FL, 1983) (Ed. P. R. Graves-Morris, E. B. Saff, and R. S. Varga). New York: Springer-Verlag, pp. 383-411, 1984.

Cody, W. J.; Meinardus, G.; and Varga, R. S. "Chebyshev Rational Approximations to in and Applications to Heat-Conduction Problems." J. Approx. Th. 2, 50-65, 1969.

Dunham, C. B. and Taylor, G. D. "Continuity of Best Reciprocal Polynomial Approximation on ." J. Approx. Th. 30, 71-79, 1980.

Finch, S. R. "The 'One-Ninth' Constant." §4.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 259-262, 2003.

Gonchar, A. A. "Rational Approximations of Analytic Functions." Amer. Math. Soc. Transl. Ser. 2 147, 25-34, 1990.

Gonchar, A. A. and Rakhmanov, E. A. "Equilibrium Distributions and Degree of Rational Approximation of Analytic Functions." Math. USSR Sbornik 62, 305-348, 1980.

Gonchar, A. A. and Rakhmanov, E. A. "Equilibrium Distributions and the Rate of Rational Approximation of Analytic Functions." Mat. Sbornik 34, 306-352, 1987. Reprinted in Math. USSR Sbornik 62, 305-348, 1989.

Magnus, A. P. "CFGT Determination of Varga's Constant '1/9'." Inst. Preprint B-1348. Belgium: Inst. Math. U.C.L., 1986.

Magnus, A. P. "On the Use of the Carathéodory-Fejér Method for Investigating '' and Similar Constants." In Nonlinear Numerical Methods and Rational Approximation (Wilrijk, 1987). Dordrecht, Netherlands: Reidel, pp. 105-132, 1988.

Rahman, Q. I. and Schmeisser, G. "Rational Approximation to the Exponential Function." In Padé and Rational Approximation, (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976) (Ed. E. B. Saff and R. S. Varga). New York: Academic Press, pp. 189-194, 1977.

Schönhage, A. "Zur rationalen Approximierbarkeit von über ." J. Approx. Th. 7, 395-398, 1973.

Sloane, N. J. A. Sequences A072558, A073007, and A086199 in "The On-Line Encyclopedia of Integer Sequences."

Trefethen, L. N. and Gutknecht, M. H. "The Caratheodory-Fejer Method for Real Rational Approximation." SINUM 20, 420-436, 1983.

Varga, R. S. Scientific Computations on Mathematical Problems and Conjectures. Philadelphia, PA: SIAM, 1990.

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Weisstein, Eric W. "One-Ninth Constant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/One-NinthConstant.html