Soldner's Constant

The logarithmic integral is defined as the Cauchy principal value

			(1)
			(2)

Soldner's constant, denoted  (or sometimes ) is the root of the logarithmic integral,

(3)

so that

(4)

for  (Soldner 1812; Nielsen 1965, p. 88). Ramanujan calculated  (Hardy 1999, Le Lionnais 1983, Berndt 1994), while the correct value is 1.45136923488... (OEIS A070769; Derbyshire 2004, p. 114).

REFERENCES:

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 123-124, 1994.

Berndt, B. C. and Evans, R. J. "Some Elegant Approximations and Asymptotic Formulas for Ramanujan." J. Comput. Appl. Math. 37, 35-41, 1991.

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.

Finch, S. R. "Euler-Gompertz Constant." §6.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 423-428, 2003.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 23 and 45, 1999.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 39, 1983.

Michon, G. P. "Final Answers: Numerical Constants." https://home.att.net/~numericana/answer/constants.htm#mertens.

Nielsen, N. "Theorie des Integrallograrithmus und Verwandter Transzendenten." Part II in Die Gammafunktion. New York: Chelsea, 1965.

Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc., p. 351, 2000.

Sloane, N. J. A. Sequence A070769 in "The On-Line Encyclopedia of Integer Sequences."

Soldner. Abhandlungen 2, 333, 1812.