One-Ninth Constant

The one-ninth constant , also called the Halphen constant (Finch 2003, p. 261), is defined by

where are Chebyshev constants. Schönhage (1973) proved that

It was conjectured that , hence the name. Its reciprocal is sometimes called Varga's constant. In 1981, N. Trefethen (Trefethen and Gutknecht 1983) refuted the conjecture that by computing Varga's constant as

(OEIS 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

(OEIS A072558) numerically.

Gonchar and Rakhmanov (1987) proved that the limit exists and disproved the conjecture. Gonchar presented the result at the International Congress of Mathematicians in Berkeley, California, in 1986 (Gonchar 1990). Magnus (1986, 1988) showed that is exactly given by

where is the complete elliptic integral of the first kind and (OEIS A086199) is the elliptic modulus in which solves

and is the complete elliptic integral of the second kind.

is also given by the unique root in of

where

and

(Gonchar and Rakhmanov 1987). may also be computed from the prime factorization , where the are distinct odd primes, , and , as

(Gonchar and Rakhmanov 1987).

A generating function for is given by

for , where and are complete elliptic integrals of the first kind and complete elliptic integrals of the 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

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.