Logarithmic Integral

The logarithmic integral is defined by

$$\mathop{\rm li}\nolimits (x) \equiv \int_0^x {du\over \ln u}.$$ (1)

The offset form appearing in the Prime Number Theorem is defined so that $\mathop{\rm Li}\nolimits (2)=0$:

$$\mathop{\rm Li}\nolimits (x) \equiv \int_2^x {du\over \ln u}$$ (2)

$$= \mathop{\rm li}\nolimits (x)-\mathop{\rm li}\nolimits (2)\approx \mathop{\rm li}\nolimits (x)-1.04516$$ (3)

$$= \mathop{\rm ei}(\ln x),$$ (4)

where $\mathop{\rm ei}\nolimits (x)$ is the Exponential Integral. (Note that the Notation $\mathop{\rm Li}\nolimits _n(z)$ is also used for the Polylogarithm.) Nielsen (1965, pp. 3 and 11) showed and Ramanujan independently discovered (Berndt 1994) that

$$\int_\mu^x {dt\over \ln t}=\gamma+\ln\ln x+\sum_{k=1}^\infty {(\ln x)^k\over k!k},$$ (5)

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

$$\int_\mu^x{dt\over\ln t}=\gamma+\ln\ln x+\sqrt{x}\sum_{n=0}^... ...}(\ln x)^n\over n!2^{n-1}}\sum_{k=0}^{[(n-1)/2]} {1\over 2k+1}$$ (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.