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Soldner's Constant


The logarithmic integral is defined as the Cauchy principal value

li(x)=PVint_0^x(dt)/(lnt)
(1)
=lim_(epsilon->0^+)[int_0^(1-epsilon)(dt)/(lnt)+int_(1+epsilon)^x(dt)/(lnt)].
(2)

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

 li(x)=0,
(3)

so that

 PVint_0^x(dt)/(lnt)=int_mu^x(dt)/(lnt)
(4)

for x>mu (Soldner 1812; Nielsen 1965, p. 88). Ramanujan calculated mu=1.45136380... (Hardy 1999, Le Lionnais 1983, Berndt 1994), while the correct value is 1.45136923488... (OEIS A070769; Derbyshire 2004, p. 114).


See also

Exponential Integral, Logarithmic Integral, Riemann Prime Counting Function, Soldner's Constant Continued Fraction, Soldner's Constant Digits

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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." http://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.

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Soldner's Constant

Cite this as:

Weisstein, Eric W. "Soldner's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SoldnersConstant.html

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