## Logarithmic Integral

The logarithmic integral is defined by

 (1)

The offset form appearing in the Prime Number Theorem is defined so that :
 (2) (3) (4)

where is the Exponential Integral. (Note that the Notation is also used for the Polylogarithm.) Nielsen (1965, pp. 3 and 11) showed and Ramanujan independently discovered (Berndt 1994) that
 (5)

where is the Euler-Mascheroni Constant and is Soldner's Constant. Another Formula due to Ramanujan which converges more rapidly is

 (6)

(Berndt 1994).

See also Polylogarithm, Prime Constellation, Prime Number Theorem, Skewes Number

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 126-131, 1994.

Nielsen, N. Theorie des Integrallogarithms. New York: Chelsea, 1965.

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 151, 1991.

© 1996-9 Eric W. Weisstein
1999-05-25